ࡱ> pQq~sghijklmnoG ybjbjَ 5]$P D9"F jJJr"9 Ex2z2z2z2=2L6LO9$:<Bs9M@MMs9}+JreF }+}+}+M\J(rx2Mx2}+}+K02hX6x2r* @[ (h2Algorithms and software for linear and nonlinear programming Stephen J. Wright Mathematics and Computer Science Division Argonne National Laboratory Argonne IL 60439 Abstract The past ten years have been a time of remarkable developments in software tools for solving optimization problems. There have been algorithmic advances in such areas as linear programming and integer programming which have now borne fruit in the form of more powerful codes. The advent of modeling languages has made the process of formulating the problem and invoking the software much easier, and the explosion in computational power of hardware has made it possible to solve large, difficult problems in a short amount of time on desktop machines. A user community that is growing rapidly in size and sophistication is driving these developments. In this article, we discuss the algorithmic state of the art and its relevance to production codes. We describe some representative software packages and modeling languages and give pointers to web sites that contain more complete information. We also mention computational servers for online solution of optimization problems. Keywords Optimization, Linear programming, Nonlinear programming, Integer programming, Software. Introduction Optimization problems arise naturally in many engineering applications. Control problems can be formulated as optimization problems in which the variables are inputs and states, and the constraints include the model equations for the plant. At successively higher levels, optimization can be used to determine setpoints for optimal operations, to design processes and plants, and to plan for future capacity. Optimization problems contain the following key ingredients: Variables that can take on a range of values. Variables that are real numbers, integers, or binary (that is, allowable values 0 and 1) are the most common types, but matrix variables are also possible. Constraints that define allowable values or scopes for the variables, or that specify relationships between the variables; An objective function that measures the desirability of a given set of variables. The optimization problem is to choose from among all variables that satisfy the constraints the set of values that minimizes the objective function. The term mathematical programming, which was coined around 1945, is synonymous with optimization. Correspondingly, linear optimization (in which the constraints and objective are linear functions of the variables) is usually known as linear programming, while optimization problems that involve constraints and have nonlinearity present in the objective or in at least some constraints, are known as nonlinear programming problems. In convex programming, the objective is a convex function and the feasible set (the set of points that satisfy the constraints) is a convex set. In quadratic programming, the objective is a quadratic function while the constraints are linear. Integer programming problems are those in which some or all of the variables are required to take on integer values. Optimization technology is traditionally made available to users by means of codes or packages for specific classes of problems. Data is communicated to the software via simple data structures and subroutine argument lists, user-written subroutines (for evaluating nonlinear objective or constraint functions), text files in the standard MPS format, or text files that describe the problem in certain vendor-specific formats. More recently, modeling languages have become an appealing way to interface to packages, as they allow the user to define the model and data in a way that makes intuitive sense in terms of the application problem. Optimization tools also form part of integrated modeling systems such as GAMS and LINDO, and even underlie spreadsheets such as Microsofts Excel. Other under the hood optimization tools are present in certain logistics packages, for example, packages for supply chain management or facility location. The majority of this paper is devoted to a discussion of software packages and libraries for linear and nonlinear programming, both freely available and proprietary. We emphasize in particular packages that have become available during the past 10 years, that address new problem areas or that make use of new algorithms. We also discuss developments in related areas such as modeling languages and automatic differentiation. Background information on algorithms and theory for linear and nonlinear programming can be found in a number of texts, including those of Luenberger (1984), Chvatal (1983), Bertsekas (1995), Nash and Sofer (1996), and the forthcoming book of Nocedal and Wright (1999). Online Resources and Computational Servers As with so many other topics, a great deal of information about optimization software is available on the world-wide web. Here we point to a few noncommercial sites that give information about optimization algorithms and software, modeling issues, and operations research. Many other interesting sites can be found by following links from the sites mentioned below. The NEOS Guide at  HYPERLINK http://www.mcs.anl.gov/otc/Guide www.mcs.anl.gov/otc/Guide contains A guide to optimization software containing around 130 entries. The guide is organized by the name of the code, and classified according to the type of problem solved by the code. An optimization tree containing a taxonomy of optimization problem types and outlines of the basic algorithms. Case studies that demonstrate the use of algorithms in solving real-world optimization problems. These include optimization of an investment portfolio, choice of a lowest-cost diet that meets a set of nutritional requirements, and optimization of a strategy for stockpiling and retailing natural gas, under conditions of uncertainty about future demand and price. The NEOS Guide also houses the FAQs for Linear and Nonlinear Programming, which can be found at  HYPERLINK http://www.mcs.anl.gov/otc/Guide/faq/ www.mcs.anl.gov/otc/Guide/faq/. These pages, updated monthly, contain basic information on modeling and algorithmic issues, information for most of the available codes in the two areas, and pointers to texts for readers who need background information. Michael Trick maintains a comprehensive web site on operations research topics at  HYPERLINK http://mat.gsia.cmu.edu http://mat.gsia.cmu.edu. It contains pointers to most online resources in operations research, together with an extensive directory of researchers and research groups and of companies that are involved in optimization and logistics software and consulting. Hans Mittelmann and Peter Spellucci maintain a decision tree to help in the selection of appropriate optimization software tools at  HYPERLINK http://plato.la.asu.edu/guide.html http://plato.la.asu.edu/guide.html. Benchmarks for a variety of codes, with an emphasis on linear programming solvers that are freely available to researchers, can be found at  HYPERLINK http://plato.la.asu.edu/bench.html http://plato.la.asu.edu/bench.html. The page  HYPERLINK http://solon.cma.univie.ac.at/~neum/glopt.html http://solon.cma.univie.ac.at/~neum/glopt.html, maintained by Arnold Neumaier, emphasizes global optimization algorithms and software. The NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server www.mcs.anl.gov/neos/Server is a computational server for the remote solution of optimization problems over the Internet. By using an email interface, a Web page, or an xwindows submission tool that connects directly to the Server via Unix sockets, users select a code and submit the model information and data that defines their problem. The job of solving the problem is allocated to one of the available workstations in the Servers pool on which that particular package is installed, then the problem is solved and the results returned to the user. The Server now has a wide variety of solvers in its roster, including a number of proprietary codes. For linear programming, the BPMPD, HOPDM, PCx, and XPRESS-MP/BARRIER interior-point codes as well as the XPRESS-MP/SIMPLEX code are available. For nonlinear programming, the roster includes LANCELOT, LOQO, MINOS, NITRO, SNOPT, and DONLP2. Input in the AMPL modeling language is accepted for many of the codes. The obvious target audience for the NEOS Server includes users who want to try out a new code, to benchmark or compare different codes on data of relevance to their own applications, or to solve small problems on an occasional basis. At a higher level, however, the Server is an experiment in using the Internet as a computational, problem-solving tool rather than simply an informational device. Instead of purchasing and installing a piece of software for installation on their local hardware, users gain access to the latest algorithmic technology (centrally maintained and updated), the hardware resources needed to execute it and, where necessary, the consulting services of the authors and maintainers of each software package. Such a means of delivering problem-solving technology to its customers is an appealing option in areas that demand access to huge amounts of computing cycles (including, perhaps, integer programming), areas in which extensive hands-on consulting services are needed, areas in which access to large, centralized, constantly changing data bases, and areas in which the solver technology is evolving rapidly. Linear Programming In linear programming problems, we minimize a linear function of real variables over a region defined by linear constraints. The problem can be expressed in standard form as where x is a vector of n real numbers, while Ax=b is a set of linear equality constraints and x>=0 indicates that all components of x are required to be nonnegative. The dual of this problem is  EMBED Equation.3  where  EMBED Equation.3 is a vector of Lagrange multipliers and  EMBED Equation.3  is a vector of dual slack variables. These two problems are intimately related, and algorithms typically solve both of them simultaneously. When the vectors  EMBED Equation.3  and  EMBED Equation.3 satisfy the following optimality conditions:  EMBED Equation.3  then  EMBED Equation.3 solves the primal problem and  EMBED Equation.3  solves the dual problem. Simple transformations can be applied to any problem with a linear objective and linear constraints (equality and inequality) to obtain this standard form. Production quality linear programming solvers carry out the necessary transformations automatically, so the user is free to specify upper bounds on some of the variables, use linear inequality constraints, and in general make use of whatever formulation is most natural for their particular application. The popularity of linear programming as an optimization paradigm stems from its direct applicability to many interesting problems, the availability of good, general-purpose algorithms, and the fact that in many real-world situations, the inexactness in the model or data means that the use of a more sophisticated nonlinear model is not warranted. In addition, linear programs do not have multiple local minima, as may be the case with nonconvex optimization problems. That is, any local solution of a linear program(one whose function value is no larger than any feasible point in its immediate vicinity(also achieves the global minimum of the objective over the whole feasible region. It remains true that more (human and computational) effort is invested in solving linear programs than in any other class of optimization problems. Prior to 1987, all of the commercial codes for solving general linear programs made use of the simplex algorithm. This algorithm, invented in the late 1940s, had fascinated optimization researchers for many years because its performance on practical problems is usually far better than the theoretical worst case. A new class of algorithms known as interior-point methods was the subject of intense theoretical and practical investigation during the period 19841995, with practical codes first appearing around 1989. These methods appeared to be faster than simplex on large problems, but the advent of a serious rival spurred significant improvements in simplex codes. Today, the relative merits of the two approaches on any given problem depend strongly on the particular geometric and algebraic properties of the problem. In general, however, good interior-point codes continue to perform as well or better than good simplex codes on larger problems when no prior information about the solution is available. When such warm start information is available, however, as is often the case in solving continuous relaxations of integer linear programs in branch-and-bound algorithms, simplex methods are able to make much better use of it than interior-point methods. Further, a number of good interior-point codes are freely available for research purposes, while the few freely available simplex codes are not quite competitive with the best commercial codes. The simplex algorithm generates a sequence of feasible iterates  EMBED Equation.3 for the primal problem, where each iterate typically has the same number of nonzero (strictly positive) components as there are rows in  EMBED Equation.3 . We use this iterate to generate dual variables  EMBED Equation.3 and  EMBED Equation.3 such that two other optimality conditions are satisfied, namely,  EMBED Equation.3  If the remaining condition  EMBED Equation.3 is also satisfied, then the solution has been found and the algorithm terminates. Otherwise, we choose one of the negative components of  EMBED Equation.3 and allow the corresponding component of  EMBED Equation.3  to increase from zero. To maintain feasibility of the equality constraint  EMBED Equation.3  the components that were strictly positive in  EMBED Equation.3 will change. One of them will become zero when we increase the new component to a sufficiently large value. When this happens, we stop and denote the new iterate by  EMBED Equation.3 . Each iteration of the simplex method is relatively inexpensive. It maintains a factorization of the submatrix of  EMBED Equation.3  that corresponds to the strictly positive components of  EMBED Equation.3 (a square matrix  EMBED Equation.3 known as the basis(and updates this factorization at each step to account for the fact that one column of  EMBED Equation.3 has changed. Typically, simplex methods converge in a number of iterates that is about two to three times the number of columns in  EMBED Equation.3 . Interior-point methods proceed quite differently, applying a Newton-like algorithm to the three equalities in the optimality conditions and taking steps that maintain strict positivity of all the  EMBED Equation.3  and  EMBED Equation.3  components. It is the latter feature that gives rise to the term interior-point (the iterates are strictly interior with respect to the inequality constraints. Each interior-point iteration is typically much more expensive than a simplex iteration, since it requires refactorization of a large matrix of the form  EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3 are diagonal matrices whose diagonal elements are the components of the current iterates  EMBED Equation.3 and  EMBED Equation.3 , respectively. The solutions to the primal and dual problems are generated simultaneously. Typically, interior-point iterates converge in between 10 and 100 iterations. Codes can differ in a number of important respects, apart from the different underlying algorithm. All practical codes include presolvers, which attempt to reduce the dimension of the problem by determining the values of some of the primal and dual variables without applying the algorithm. As a simplex example, suppose that the linear program contains the constraints  EMBED Equation.3  then the only possible values for the three variables are  EMBED Equation.3  These variables can be fixed and deleted from the problem, along with the three corresponding columns of  EMBED Equation.3 and the three components of  EMBED Equation.3 . Presolve techniques have become quite sophisticated over the years, though little has been written about them because of their commercial value. An exception is the paper of Andersen and Andersen (1995). For information on specific codes, refer to the online resources listed above, in particular, the NEOS Software Guide, the Linear Programming FAQ, and the benchmarks maintained by Hans Mittelmann. Modern, widely used commercial simplex codes include CPLEX and the XPRESS-MP. Both these codes accept input in the industry-standard MPS format, and also in their own proprietary formats. Both have interfaces to various modeling languages, and also a callable library interface that allows users to set up, modify, and solve problems by means of function calls from C or FORTRAN code. Both packages are undergoing continual development. Freely available simplex codes are usually of lower quality, with the exception of SOPLEX. This is a C++ code written as a thesis project by Roland Wunderling, can be found at www.zib.de/Optimization/Software/Soplex/. The code MINOS is available to nonprofit and academic researchers for a nominal fee. Commercial interior-point solvers are available as options in the CPLEX and XPRESS-MP packages. However, a number of highly competitive codes are available free for research and noncommercial use, and can for the most part be obtained through the Web. Among these are BPMPD, PCx, COPLLP, LOQO, HOPDM, and LIPSOL. See Mittelmanns benchmark page for comparisons of these code and links to their web sites. These codes mostly charge a license fee for commercial use, but it is typically lower than for fully commercial packages. All can read MPS files, and most are interfaced to modeling languages. LIPSOL is programmed in Matlab (with the exception of the linear equations solver), while the other codes are written in C and/or FORTRAN. A fine reference on linear programming, with an emphasis on the simplex method, is the book of Chvatal (1983). An online Java applet that demonstrates the operation of the simplex method on small user-defined problems can be found at www.mcs.anl.gov/otc/Guide/CaseStudies/simplex/. Wright (1997) gives a description of practical interior-point methods. Modeling Languages From the users point of view, the efficiency of the algorithm or the quality of the programming may not be the critical factors in determining the usefulness of the code. Rather, the ease with which it can be interfaced to his particular applications may be more important; weeks of person-hours may be more costly to the enterprise than a few hours of time on a computer. The most suitable interface depends strongly on the particular application and on the context in which it is solved. For users that are well acquainted with a spreadsheet interface, for instance, or with MATLAB, a code that can accept input from these sources may be invaluable. For users with large legacy modeling codes that set up and solve optimization problems by means of subroutine calls, substitution of a more efficient package that uses more or less the same subroutine interface may be the best option. In some disciplines, (JPs biology/chemistry pointer) application-specific modeling languages allow problems to be posed in a thoroughly intuitive way. In other cases, application-specific graphical user interfaces may be more appropriate. For general optimization problems, a number of high-level modeling languages have become available that allow problems to be specified in intuitive terms, using data structures, naming schemes, and algebraic relational expressions that are dictated by the application and model rather than by the input requirements of the optimization code. Typically, a user starting from scratch will find the process of model building more straightforward and bug free with such a modeling language than, say, a process of writing FORTRAN code to pack the data into one-dimensional arrays, turning the algebraic relations between the variables into FORTRAN expressions involving elements of these arrays, and writing more code to interpret the output from the optimization routine in terms of the original application. The following simple example in AMPL demonstrates the usefulness of a modeling language (see Fourer, Gay, and Kernighan (1993), page 11). The application is to a steel production model, in which the aim is to maximize profit obtained from manufacturing a number of steel products by choosing the amount of each product to manufacture, subject to restrictions on the maximum demands for each product and the time available in each work week to manufacture them. The following file is an AMPL model file that specifies the variables, the parameters that quantify aspects of the model, and the constraints and objective. set PROD; param rate {PROD} >0; param avail >= 0; param profit {PROD}; param market{PROD}; var Make {p in PROD} >= 0, <= market[p]; maximize total_profit: sum {p in PROD} profit[p] *Make[p]; subject to Time: sum {p in PROD} (1/rate[p]) * Make[p] <= avail; PROD is the collection of possible products that can be manufactured, while rate, profit and market are the rate at which each product can be manufactured, the profit on each product, and the maximum demand for each product, respectively. avail represents the total time available for manufacturing. Make is the variable in the problem, representing the amount of each product to be manufactured. In its definition, each element of Make is constrained to lie between zero and the maximum demand for the product in question. The last two lines of the mode file specify the objective and constraint in a self-evident fashion. The actual values of the parameters can be assigned by means of additional statements in this file, or in a separate data file. For instance, the following data file specifies parameters for two products, bands and coils: set PROD := bands coils; param: rate profit market := bands 200 25 6000 coils 140 30 4000; param avail := 40; These statements specify that the market[bands] is 6000, profit[bands] is 25, and so on. An interactive AMPL session would proceed by invoking commands to read these two files and then invoking an option solver command to choose the linear programming solver to be used (for example, CPLEX or PCx) together with settings for parameters such as stopping tolerances, etc, that the user may wish to change from their defaults. A solve command would then solve the problem (and report messages passed through from the underlying optimization code). Results can be inspected by invoking the display command. For the above example, the command display Make invoked after the problem has been solved would produce the following output: Make [*] := bands 6000 coils 0 ; Note from this example the intuitive nature of the algebraic relations, and the fact that we could index the parameter arrays by the indices bands and coils, rather than the numerical indices 1 and 2 that would be required if we were programming in FORTRAN. Note too that additional products can be added to the mix without changing the model file at all. Of course, the features of AMPL are much more extensive than the simple example above allows us to demonstrate. The web site  HYPERLINK http://www.ampl.com www.ampl.com contains a great deal of information about the language and the optimization software to which it is linked, and allows users to solve their own simple models online. Numerous other modeling languages and systems can be found on the online resources described above, particularly the NEOS Software Guide and the linear and nonlinear programming FAQs. We mention in particular AIMMS (Bisschop and Entriken (1993)) which has a built in graphical interface; GAMS, a well established system available with support for linear, nonlinear, and mixed-integer programming and newly added procedural featured; and MPL, a Windows-based system whose web site  HYPERLINK http://www.maximal-usa.com www.maximal-usa.com contains a comprehensive tutorial and a free student version of the language. Other Input Formats The established input format for linear programming problems has from the earliest days been MPS, a column oriented format (well suited to 1950s card readers) in which names are assigned to each primal and dual variable, and the data elements that define the problem are assigned in turn. Test problems for linear programming are still distributed in this format. It has significant disadvantages, however. The format is non-intuitive and the files are difficult to modify. Moreover, it restricts the precision to which numerical values can be specified. The format survives only because no universally accepted standard has yet been developed to take its place. As mentioned above, vendors such as CPLEX and XPRESS have their own input formats, which avoid the pitfalls of MPS. These formats lack the portability of the modeling languages described above, but they come bundled with the code, and may be attractive for users willing to make a commitment to a single vendor. For nonlinear programming, SIF (the standard input format) was proposed by the authors of the LANCELOT code in the early 1990s. SIF is somewhat hamstrung by the fact that it is compatible with MPS. SIF files have a similar look to MPS files, except that there are a variety of new keywords for defining variables, groups of variables, and the algebraic relationships between them. For developers of nonlinear programming software, SIF has the advantage that a large collection of test problems(the CUTE test set(is available in this format. For users, however, formulating a model in SIF is typically much more difficult than using one of the modeling languages of the previous section. For complete information about SIF, see http://www.numerical.rl.ac.uk/lancelot/sif/sifhtml.html Nonlinear Programming Nonlinear programming problems are constrained optimization problems with nonlinear objective and/or constraint functions. However, and we still assume that all functions in question are smooth (typically, at least twice differentiable), and that the variables are all real numbers. If any of the variables are required to take on integer values, the problem is a (mixed-) integer nonlinear programming problem, a class that we will not consider in this paper. For purposes of description, we use the following formulation of the problem:  EMBED Equation.3 , where  EMBED Equation.3 is a vector of  EMBED Equation.3 real variables,  EMBED Equation.3 is a smooth real-valued function, and  EMBED Equation.3 and  EMBED Equation.3 are smooth functions with dimension  EMBED Equation.3 and  EMBED Equation.3 , respectively. Algorithms for nonlinear programming problems are more varied that those for linear programming. The major approaches represented in production software packages are sequential quadratic programming, reduced gradient, sequential linearly constrained, and augmented Lagrangian methods. (The latter is also known as the method of multipliers.) Extension of the successful interior-point approaches for linear programming to the nonlinear problem is the subject of intense ongoing investigation among optimization researchers, but little production software for these approaches is yet available. The use of nonlinear models may be essential in some applications, since a linear or quadratic model may be too simplistic and therefore produce useless results. However there is a price to pay for using the more general nonlinear paradigm. For one thing, most algorithms cannot guarantee convergence to the global minimum, i.e., the value EMBED Equation.3 that minimizes EMBED Equation.3 over the entire feasible region. At best, they will find a point that yields the smallest value of  EMBED Equation.3 over all points in some feasible neighborhood of itself. (An exception occurs in convex programming, in which the functions  EMBED Equation.3 and  EMBED Equation.3 are convex, while  EMBED Equation.3 are linear. In this case, any local minimizer is also a global minimizer. Note that linear programming is a special case of convex programming.) The problem of finding the global minimizer, though an extremely important one in some applications such as molecular structure determination, is very difficult to solve. While several general algorithmic approaches for global optimization are available, they are invariably implemented in a way that exploits heavily the special properties of the underlying application, so that there is a fair chance that they will produce useful results in a reasonable amount of computing time. We refer to Floudas and Pardalos (1992) and the journal Global Optimization for information on recent advances in this area. A second disadvantage of nonlinear programming over linear programming is that general-purpose software is somewhat less effective, because the nonlinear paradigm encompasses such a wide range of problems with a great number of potential pathologies and eccentricities. Even when we are close to a minimizer  EMBED Equation.3 , algorithms may encounter difficulties because the solution may be degenerate, in the sense that certain of the active constraints become dependent, or are only weakly active. Curvature in the objective or constraint functions (a second-order effect not present in linear programming), and differences in this curvature between different directions, can cause difficulties for the algorithms, especially when second derivative information is not supplied by the user or not exploited by the algorithm. Finally, some of the codes treat the derivative matrices as dense, which means that they the maximum dimension of the problems they can handle is somewhat limited. However, most of the leading codes, including LANCELOT, MINOS, SNOPT, and SPRNLP are able to exploit sparsity, and are therefore equipped to handle large-scale problems. Algorithms for special cases of the nonlinear programming problem, such as problems in which all constraints are linear or the only constraints are bounds on the components of  EMBED Equation.3 , tend to be more effective than algorithms for the general problem because they are more able to exploit the special properties. (We discuss a few such special cases below.) Even for problems in which the constraints are nonlinear, the problem may contain special structure that can be exploited by the algorithm or by the routines that perform linear algebra operations at each iteration. An example is the optimal control problem (arising, for example, in model predictive control), in which the equality constraint represents a nonlinear model of the plant, and the inequalities represent bounds and other restrictions on the states and inputs. The Jacobian (matrix of first partial derivatives of the constraints) typically has a banded structure, while the Hessian of the objective is symmetric and banded. Linear algebra routines that exploit this bandedness, or dig even deeper and exploit the control origins of the problem, are much more effective than general routines on such problems. Local solutions of the nonlinear program can be characterized by a set of optimality conditions analogous to those described above for the linear programming problem. We introduce Lagrange multipliers  EMBED Equation.3  and  EMBED Equation.3 for the constraints  EMBED Equation.3  and  EMBED Equation.3 , respectively, and write the Lagrangian function for this problem as  EMBED Equation.3  The first-order optimality conditions (commonly known as the KKT conditions) are satisfied at a point  EMBED Equation.3  if there exist multiplier vectors  EMBED Equation.3  and  EMBED Equation.3  such that  EMBED Equation.3  The active constraints are those for while equality holds at  EMBED Equation.3 . All the components of  EMBED Equation.3  are active by definition, while the active components of  EMBED Equation.3  are those for which  EMBED Equation.3  When the constraint gradients satisfy certain regularity conditions at  EMBED Equation.3 , the KKT conditions are necessary for  EMBED Equation.3  to be a local minimizer of the nonlinear program, but not sufficient. A second-order sufficient condition is that the Hessian of the Lagrangian, the matrix  EMBED Equation.3 , has positive curvature along all directions that lie in the null space of the active constraint gradients, for some choice of multipliers  EMBED Equation.3  and  EMBED Equation.3  that satisfy the KKT conditions. That is, we require  EMBED Equation.3  for all vectors  EMBED Equation.3  such that  EMBED Equation.3  and  EMBED Equation.3  for all active indices  EMBED Equation.3 . The sequential quadratic programming (SQP) approach has been investigated extensively from a theoretical point of view and is the basis of several important practical codes, including NPSOL and the more recent SNOPT. It works by approximating the nonlinear programming problem by a quadratic program around the current iterate  EMBED Equation.3 , that is,  EMBED Equation.3  where EMBED Equation.3 is a symmetric matrix (usually positive definite) that contains exact or approximate second-order information about the objective and constraint functions. There are many modifications of this basic scheme. For instance, a trust-region bound limiting the length of the step  EMBED Equation.3 may be added to the model, or the linear constraints may be adjusted so that the current step is not required to remedy all the infeasibility in the current iterate  EMBED Equation.3 . The approximate Hessian  EMBED Equation.3  can be chosen in a number of ways. Local quadratic convergence can be proved under certain assumptions if this matrix is set to the Hessian of the Lagrangian, that is,  EMBED Equation.3 evaluated at the primal iterate EMBED Equation.3 and the current estimates  EMBED Equation.3  of the Lagrange multiplier vectors. The code SPRNLP allows users to select this value for  EMBED Equation.3 , provided that they are willing to supply the second derivative information. Alternatively,  EMBED Equation.3  can be a quasi-Newton approximation to the Lagrangian Hessian. Update strategies that yield local superlinear convergence are well known, and are implemented in dense codes such as NPSOL, DONLP2, NLPQL, and are available as an option in a version of SPRNLP that does not exploit sparsity. SNOPT also uses quasi-Newton Hessian approximations, but unlike the codes just mentioned it is able to exploit sparsity and is therefore better suited to large-scale problems. Another quasi-Newton variant is to maintain an approximation to the reduced Hessian, the two-sided projection of this matrix onto the null space of the active constraints. The latter approach is particularly efficient when the dimension of this null space is small in relation to the number of components of  EMBED Equation.3 , as is the case in many process control problems, for instance. The approach does not appear to be implemented in general-purpose SQP software, however. To ensure that the algorithm converges to a point satisfying the KKT conditions from any starting point, the basic SQP algorithm must be enhanced by the addition of a global convergence strategy. Usually, this strategy involves a merit function, whose purposes is to evaluate the desirability of a given iterate  EMBED Equation.3  by accounting for its objective value and the amount by which it violates the constraints. The commonly used  EMBED Equation.3  penalty function simply forms a weighted average of the objective and the constraint violations, as follows:  EMBED Equation.3  where  EMBED Equation.3  is the vector of length  EMBED Equation.3  whose elements are  EMBED Equation.3 and  EMBED Equation.3  is a positive parameter. The simplest algorithm based on this function fixes  EMBED Equation.3  and insists that all steps produce a sufficient decrease in the value of  EMBED Equation.3 . Line search or trust region strategies are applied to ensure that steps with the required property can be found whenever the current point  EMBED Equation.3  does not satisfy the KKT conditions. More sophisticated strategies contain mechanisms for adjusting the parameter  EMBED Equation.3  and for ensuring that the fast local convergence properties are not compromised by the global convergence strategy. We note that the terminology can be confusing(global convergence in this context refers to convergence to a KKT point from any starting point, and not to convergence to a global minimizer. For more information on SQP, we refer to the review paper of Boggs and Tolle (1996), and Chapter 18 of Nocedal and Wright (1999). A second algorithmic approach is known variously as the augmented Lagrangian method or the method of multipliers. Noting that the first KKT condition, namely,  EMBED Equation.3 , requires EMBED Equation.3 to be a stationary point of the Lagrangian function  EMBED Equation.3 , we modify this function to obtain an augmented function for which  EMBED Equation.3  is not just a stationary point but also a minimizer. When only equality constraints are present (that is,  EMBED Equation.3  is vacuous), the augmented Lagrangian function has the form  EMBED Equation.3  where  EMBED Equation.3  is a positive parameter. It is not difficult to show that if  EMBED Equation.3 is set to its optimal value  EMBED Equation.3  (the value that satisfies the KKT conditions) and  EMBED Equation.3  is sufficiently large, that  EMBED Equation.3  is a minimizer of  EMBED Equation.3 . Intuitively, the purpose of the squared-norm term is to add positive curvature to the function  EMBED Equation.3  in just those directions in which it is needed(the directions in the range space of the active constraint gradients. (We know already from the second-order sufficient conditions that the curvature of  EMBED Equation.3  in the null space of the active constraint gradients is positive.) In the augmented Lagrangian method, we exploit this property by alternating between steps of two types: Fixing  EMBED Equation.3  and  EMBED Equation.3 , and finding the value of  EMBED Equation.3  that approximately minimizes  EMBED Equation.3 ; Updating  EMBED Equation.3  to make it a better approximation to  EMBED Equation.3 . The update formula for  EMBED Equation.3  has the form  EMBED Equation.3  where  EMBED Equation.3  is the approximate minimizing value just calculated. Simple constraints such as bounds or linear equalities can be treated explicitly in the subproblem, rather than included in the second and third terms of  EMBED Equation.3 . (In LANCELOT, bounds on components of  EMBED Equation.3  are treated in this manner.) Practical augmented Lagrangian algorithms also contain mechanisms for adjusting the parameter EMBED Equation.3  and for replacing the squared norm term  EMBED Equation.3  by a weighted norm that more properly reflects the scaling of the constraints and their violations at the current point. When inequality constraints are present in the problem, the augmented Lagrangian takes on a slightly more complicated form that is nonetheless not difficult to motivate. We define the function  EMBED Equation.3 as follows:  EMBED Equation.3  The definition of  EMBED Equation.3  is then modified to incorporate the inequality constraints as follows:  EMBED Equation.3  The update formula for the approximate multipliers  EMBED Equation.3  is  EMBED Equation.3  See the references below for details on derivation of this form of the augmented Lagrangian. The definitive implementation of the augmented Lagrangian approach for general-purpose nonlinear programming problems is LANCELOT. It incorporates sparse linear algebra techniques, including preconditioned iterative linear solvers, making it suitable for large-scale problems. The subproblem of minimizing the augmented Lagrangian with respect to  EMBED Equation.3  is a bound-constrained minimization problem, which is solved by an enhanced gradient projection technique. Problems can be passed to Lancelot via subroutine calls, SIF input files, and AMPL. For theoretical background on the augmented Lagrangian approach, consult the books of Bertsekas (1982, 1995), and by Conn, Gould, and Toint (1992), the authors of LANCELOT. The latter book is notable mainly for its pointers to the papers of the same three authors in which the theory of Lancelot is developed. A brief derivation of the theory appears in Chapter 17 of Nocedal and Wright (1999). (Note that the inequality constraints in this reference are assumed to have the form  EMBED Equation.3  rather than  EMBED Equation.3 , necessitating a number of sign changes in the analysis.) Interior-point solvers for nonlinear programming are the subjects of intense current investigation. An algorithm of this class, known as the sequential unconstrained minimization technique (SUMT) was actually proposed in the 1960s, in the book of Fiacco and McCormick (1968). The idea at that time was to define a barrier-penalty function for the NLP as follows:  EMBED Equation.3 where  EMBED Equation.3  is a small positive parameter. Given some value of  EMBED Equation.3 , the algorithm finds an approximation to the minimizer  EMBED Equation.3  of  EMBED Equation.3 . It then decreases  EMBED Equation.3  and repeats the minimization process. Under certain assumptions, one can show that  EMBED Equation.3  as  EMBED Equation.3  so the sequence of iterates generated by SUMT should approach the solution of the nonlinear program provided that  EMBED Equation.3  is decreased to zero. The difficulties with this approach are that all iterates must remain strictly feasible with respect to the inequality constraints (otherwise the log functions are not defined), and the subproblem of minimizing  EMBED Equation.3  becomes increasingly difficult to solve as  EMBED Equation.3  becomes small, as the Hessian of this function becomes highly ill conditioned and the radius of convergence becomes tiny. Many implementations of this approach were attempted, including some with enhancements such as extrapolation to obtain good starting points for each value of  EMBED Equation.3 . However, the approach does not survive in the present generation of software, except through its profound influence on the interior-point research of the past 15 years. Some algorithms for nonlinear programming that have been proposed in recent years contain echoes of the barrier function  EMBED Equation.3 , however. For instance, the NITRO algorithm (Byrd, Gilbert, and Nocedal (1996)) reformulates the subproblem for a given positive value of  EMBED Equation.3  as follows:  EMBED Equation.3  NITRO then applies a trust-region SQP algorithm for equality constrained optimization to this problem, choosing the trust region to have the form  EMBED Equation.3  where the diagonal matrix EMBED Equation.3  and the trust-region radius  EMBED Equation.3  are chosen so that the step  EMBED Equation.3  does not violate strict positivity of the  EMBED Equation.3  components, that is,  EMBED Equation.3 NITRO is available through the NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server/ www.mcs.anl.gov/neos/Server/ . The user is required to specify the problem by means of FORTRAN subroutines to evaluate the objective and constraints. Derivatives are obtained automatically by means of ADIFOR. An alternative interior-point approach is closer in spirit to the successful primal-dual class of linear programming algorithms. These methods generate iterates by applying Newton-like methods to the equalities in the KKT conditions. After introducing the slack variables  EMBED Equation.3  for the inequality constraints, we can restate the KKT conditions as follows:  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3 are diagonal matrices formed from the vectors  EMBED Equation.3  and  EMBED Equation.3 , respectively, while  EMBED Equation.3  is the vector  EMBED Equation.3 . We generate a sequence of iterates  EMBED Equation.3  satisfying the strict inequality  EMBED Equation.3  by applying Newton-like method to the system of nonlinear equations formed by the first four conditions above. Modification of this basic approach to ensure global convergence is the major challenge associated with this class of solvers; the local convergence theory is relatively well understood. Merit functions can be used, along with line searches and modifications to the matrix in the equations that are solved for each step, to ensure that each step at least produces a decrease in the merit function. However, no fully satisfying complete theory has yet been proposed. The code LOQO implements a primal-dual approach for nonlinear programming problems. It requires the problem to be specified in AMPL, whose built-in automatic differentiation features are used to obtain the derivatives of the objective and constraints. LOQO is also available through the NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server/ www.mcs.anl.gov/neos/Server/ , and or can be obtained for a variety of platforms. The reduced gradient approach has been implemented in several codes that have been available for some years, notably, CONOPT and LSGRG2. This approach uses the formulation in which only bounds and equality constraints are present. (Any nonlinear program can be transformed to this form by introducing slacks for the inequality constraints and constraining the slacks to be nonnegative.) Reduced gradient algorithms partition the components of  EMBED Equation.3  into three classes: basic, fixed, and superbasic. The equality constraint  EMBED Equation.3  is used to eliminate the basic components from the problem by expressing them implicitly in terms of the fixed and superbasic components. The fixed components are those that are fixed at one of their bounds for the current iteration. The superbasics are the components that are allowed to move in a direction that reduces the value of the objective  EMBED Equation.3 . Strategies for choosing this direction are derived from unconstrained optimization; they include steepest descent, nonlinear conjugate gradient, and quasi-Newton strategies. Both CONOPT and LSGRG2 use sparse linear algebra techniques during the elimination of the basic components, making them suitable for large-scale problems. While these codes have found use in many engineering applications, their performance is often slower than competing codes based on SQP and augmented Lagrangian algorithms. Finally, we mention MINOS, a code that has been available for many years in a succession of releases, and that has proved its worth in a great many engineering applications. When the constraints are linear, MINOS uses a reduced gradient algorithm, maintaining feasibility at all iterations and choosing the superbasic search direction with a quasi-Newton technique. When nonlinear constraints are present, MINOS forms linear approximations to them and replaces the objective with a projected augmented Lagrangian function in which the deviation from linearity is penalized. Convergence theory for this approach is not well established(the author admits that a reliable merit function is not known(but it appears to converge on most problems. The NEOS Guide page for SNOPT contains some guidance for users who are unsure whether to use MINOS or SNOPT. It describes problem features that are particularly suited to each of the two codes. Obtaining Derivatives One onerous requirement of some nonlinear programming codes has been their requirement that the user supply code for calculating derivatives of the objective and constraint functions. An important development of the past 10 years is that this requirement has largely disappeared. Modeling languages such as AMPL contain their own built-in systems for calculating first derivatives at specified values of the variable vector  EMBED Equation.3 , and supplying them to the underlying optimization code on request. Automatic differentiation software tools such as ADIFOR (Bischof et al. (1996)), which works with FORTRAN code, have been used to obtain derivatives from extremely complex dusty deck function evaluation routines. In the NEOS Server, all of the nonlinear optimization routines (including LANCELOT, SNOPT, and NITRO) are linked to ADIFOR, so that the user needs only to supply FORTRAN code to evaluate the objective and constraint functions, not their derivatives. Other high quality software tools for automatic differentiation include ADOL-C (Griewank, Juedes, and Utke (1996)), ODYSSEE (Rostaing, Dalmas, and Galligo (1993)), and ADIC (Bischof, Roh, and Mauer (1997)). References Andersen, E. D. and Andersen, K. D. (1995). Presolving in linear programming. Math. Prog., 71, 221-245. Bertsekas, D. P. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York. Bertsekas, D. P. (1995). Nonlinear Programming. Athena Scientific. Bischof, C., Carle, A., Khademi, P., and Mauer, A. (1996). ADIFOR 2.0: Automatic differentiation of FORTRAN programs. IEEE Computational Science and Engineering, 3, 18-32. Bischof, C., Roh, L., and Mauer, A. (1997). ADIC: An extensible automatic differentiation tool for ANSI-C. Software-Practice and Experience, 27, 1427-1456. Bisschop, J. and Entriken, R. (1993). AIMMS: The Modeling System. Available from AIMMS web site at http://www.paragon.nl Boggs, P. T. and Tolle, J. W. (1996). Sequential quadratic programming, Acta Numerica, 4, 1-51. Byrd, R. H., Gilbert, J.-C., and Nocedal, J. (1996). A trust-region algorithm based on interior-point techniques for nonlinear programming. OTC Technical Report 98/06, Optimization Technology Center. (Revised, 1998.) Chvatal, V. (1983). Linear Programming. Freeman, New York. Conn, A. R., Gould, N. I. M., and Toint, Ph. L. (1992). LANCELOT: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A). Volume 17, Springer Series in Computational Mathematics, Springer-Verlag, New York. Czyzyk, J., Mesnier. M. P., and More, J. J. (1998). The NEOS Server. IEEE Journal on Computational Science and Engineering, 5, 68-75. Fiacco, A. V. and McCormick, G. P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Tecnhiques. John Wiley and Sons, New York. (Reprinted by SIAM Publications, 1990.) Floudas, C. and Pardalos, P., eds (1992). Recent Advances in Global Optimization. Princeton University Press, Princeton. Fourer, R., Gay, D. M., and Kernighan, B. W. (1993). AMPL: A Modeling Language for Mathematical Programming.The Scientific Press, San Francisco. Griewank, A., Juedes, D., and Utke, J. (1996). ADOL-C, A package for the automatic differentiation of algorithms written in C/C++. ACM Transactions on Mathematical Software, 22, 131-167. Luenberger, D. (1984). Introduction to Linear and Nonlinear Programming. Addison Wesley. Nash, S. and Sofer, A. (1996). Linear and Nonlinear Programming. McGraw-Hill. Nocedal, J. and Wright, S. J. (forthcoming,1999). Numerical Optimization. Springer, New York. Rostaing, N., Dalmas, S., and Galligo, A. (1993). Automatic differentiation in Odyssee. Tellus, 45a, 558-568. Wright, S. J. (1997). Primal-Dual Interior-Point Methods. 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Wright Steve Wright  FMicrosoft Word Document MSWordDocWord.Document.89q+,-ABC[\tuvwxy8ċ( !&  !$\&&  !8\&qrsyBDFxzrtֹعڹz|¼ļƼҿԿ024xz<>TV*,bdfbdprj|; B*UVmH6 jUmHjXUjVUjUUjTUjTUjSU0J+jRU jU jPEHU@@penR~&F0[" |>A1na 8 $8_0~_ cgKRKzXQMe`UOȽ8B AI! \8;8+F&&\<0` $`Algorithms and software for linear and nonlinear programming Stephen J. Wright Mathematics and Computer Science Division Argonne National Laboratory Argonne IL 60439 Abstract The past ten years have been a time of remarkable developments in software tools for solving optimization problems. There have been algorithmic advances in such areas as linear programming and integer programming which have now borne fruit in the form of more powerful codes. The advent of modeling languages has made the process of formulating the problem and invoking the software much easier, and the explosion in computational power of hardware has made it possible to solve large, difficult problems in a short amount of time on desktop machines. A user community that is growing rapidly in size and sophistication is driving these developments. In this article, we discuss the algorithmic state of the art and its relevance to production codes. We describe some representative software packages and modeling languages and give pointers to web sites that contain more complete information. We also mention computational servers for online solution of optimization problems. Keywords Optimization, Linear programming, Nonlinear programming, Integer programming, Software. Introduction Optimization problems arise naturally in many engineering applications. Control problems can be formulated as optimization problems in which the variables are inputs and states, and the constraints include the model equations for the plant. At successively higher levels, optimization can be used to determine setpoints for optimal operations, to design processes and plants, and to plan for future capacity. Optimization problems contain the following key ingredients: Variables that can take on a range of values. Variables that are real numbers, integers, or binary (that is, allowable values 0 and 1) are the most common types, but matrix variables are also possible. Constraints that define allowable values or scopes for the variables, or that specify relationships between the variables; An objective function that measures the desirability of a given set of variables. The optimization problem is to choose from among all variables that satisfy the constraints the set of values that minimizes the objective function. The term  mathematical programming , which was coined around 1945, is synonymous with optimization. Correspondingly, linear optimization (in which the constraints and objective are linear functions of the variables) is usually known as  linear programming, while optimization problems that involve constraints and have nonlinearity present in the objective or in at least some constraints, are known as  nonlinear programming problems. In convex programming, the objective is a convex function and the feasible set (the set of points that satisfy the constraints) is a convex set. In quadratic programming, the objective is a quadratic function while the constraints are linear. Integer programming problems are those in which some or all of the variables are required to take on integer values. Optimization technology is traditionally made available to users by means of codes or packages for specific classes of problems. Data is communicated to the software via simple data structures and subroutine argument lists, user-written subroutines (for evaluating nonlinear objective or constraint functions), text files in the standard MPS format, or text files that describe the problem in certain vendor-specific formats. More recently, modeling languages have become an appealing way to interface to packages, as they allow the user to define the model and data in a way that makes intuitive sense in terms of the application problem. Optimization tools also form part of integrated modeling systems such as GAMS and LINDO, and even underlie spreadsheets such as Microsoft s Excel. Other  under the hood optimization tools are present in certain logistics packages, for example, packages for supply chain management or facility location. The majority of this paper is devoted to a discussion of software packages and libraries for linear and nonlinear programming, both freely available and proprietary. We emphasize in particular packages that have become available during the past 10 years, that address new problem areas or that make use of new algorithms. We also discuss developments in related areas such as modeling languages and automatic differentiation. Background information on algorithms and theory for linear and nonlinear programming can be found in a number of texts, including those of Luenberger (1984), Chvatal (1983), Bertsekas (1995), Nash and Sofer (1996), and the forthcoming book of Nocedal and Wright (1999). Online Resources and Computational Servers As with so many other topics, a great deal of information about optimization software is available on the world-wide web. Here we point to a few noncommercial sites that give information about optimization algorithms and software, modeling issues, and operations research. Many other interesting sites can be found by following links from the sites mentioned below. The NEOS Guide at  HYPERLINK http://www.mcs.anl.gov/otc/Guide www.mcs.anl.gov/otc/Guide contains A guide to optimization software containing around 130 entries. The guide is organized by the name of the code, and classified according to the type of problem solved by the code. An  optimization tree containing a taxonomy of optimization problem types and outlines of the basic algorithms. Case studies that demonstrate the use of algorithms in solving real-world optimization problems. These include optimization of an investment portfolio, choice of a lowest-cost diet that meets a set of nutritional requirements, and optimization of a strategy for stockpiling and retailing natural gas, under conditions of uncertainty about future demand and price. The NEOS Guide also houses the FAQs for Linear and Nonlinear Programming, which can be found at  HYPERLINK http://www.mcs.anl.gov/otc/Guide/faq/ www.mcs.anl.gov/otc/Guide/faq/. These pages, updated monthly, contain basic information on modeling and algorithmic issues, information for most of the available codes in the two areas, and pointers to texts for readers who need background information. Michael Trick maintains a comprehensive web site on operations research topics at  HYPERLINK http://mat.gsia.cmu.edu http://mat.gsia.cmu.edu. It contains pointers to most online resources in operations research, together with an extensive directory of researchers and research groups and of companies that are involved in optimization and logistics software and consulting. Hans Mittelmann and Peter Spellucci maintain a decision tree to help in the selection of appropriate optimization software tools at  HYPERLINK http://plato.la.asu.edu/guide.html http://plato.la.asu.edu/guide.html. Benchmarks for a variety of codes, with an emphasis on linear programming solvers that are freely available to researchers, can be found at  HYPERLINK http://plato.la.asu.edu/bench.html http://plato.la.asu.edu/bench.html. The page  HYPERLINK http://solon.cma.univie.ac.at/~neum/glopt.html http://solon.cma.univie.ac.at/~neum/glopt.html, maintained by Arnold Neumaier, emphasizes global optimization algorithms and software. The NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server www.mcs.anl.gov/neos/Server is a computational server for the remote solution of optimization problems over the Internet. By using an email interface, a Web page, or an xwindows  submission tool that connects directly to the Server via Unix sockets, users select a code and submit the model information and data that define their problem. The job of solving the problem is allocated to one of the available workstations in the Server s pool on which that particular package is installed, then the problem is solved and the results returned to the user. The Server now has a wide variety of solvers in its roster, including a number of proprietary codes. For linear programming, the BPMPD, HOPDM, PCx, and XPRESS-MP/BARRIER interior-point codes as well as the XPRESS-MP/SIMPLEX code are available. For nonlinear programming, the roster includes LANCELOT, LOQO, MINOS, NITRO, SNOPT, and DONLP2. Input in the AMPL modeling language is accepted for many of the codes. The obvious target audience for the NEOS Server includes users who want to try out a new code, to benchmark or compare different codes on data of relevance to their own applications, or to solve small problems on an occasional basis. At a higher level, however, the Server is an experiment in using the Internet as a computational, problem-solving tool rather than simply an informational device. Instead of purchasing and installing a piece of software for installation on their local hardware, users gain access to the latest algorithmic technology (centrally maintained and updated), the hardware resources needed to execute it and, where necessary, the consulting services of the authors and maintainers of each software package. Such a means of delivering problem-solving technology to its customers is an appealing option in areas that demand access to huge amounts of computing cycles (including, perhaps, integer programming), areas in which extensive hands-on consulting services are needed, areas in which access to large, centralized, constantly changing data bases, and areas in which the solver technology is evolving rapidly. Linear Programming In linear programming problems, we minimize a linear function of real variables over a region defined by linear constraints. The problem can be expressed in standard form as where x is a vector of n real numbers, while  EMBED Equation.3 is a set of linear equality constraints and  EMBED Equation.3  indicates that all components of x are required to be nonnegative. The dual of this problem is  EMBED Equation.3  where  EMBED Equation.3 is a vector of Lagrange multipliers and  EMBED Equation.3  is a vector of dual slack variables. These two problems are intimately related, and algorithms typically solve both of them simultaneously. When the vectors  EMBED Equation.3  and  EMBED Equation.3 satisfy the following optimality conditions:  EMBED Equation.3  then  EMBED Equation.3 solves the primal problem and  EMBED Equation.3  solves the dual problem. Simple transformations can be applied to any problem with a linear objective and linear constraints (equality and inequality) to obtain this standard form. Production quality linear programming solvers carry out the necessary transformations automatically, so the user is free to specify upper bounds on some of the variables, use linear inequality constraints, and in general make use of whatever formulation is most natural for their particular application. The popularity of linear programming as an optimization paradigm stems from its direct applicability to many interesting problems, the availability of good, general-purpose algorithms, and the fact that in many real-world situations, the inexactness in the model or data means that the use of a more sophisticated nonlinear model is not warranted. In addition, linear programs do not have multiple local minima, as may be the case with nonconvex optimization problems. That is, any local solution of a linear program(one whose function value is no larger than any feasible point in its immediate vicinity(also achieves the global minimum of the objective over the whole feasible region. It remains true that more (human and computational) effort is invested in solving linear programs than in any other class of optimization problems. Prior to 1987, all of the commercial codes for solving general linear programs made use of the simplex algorithm. This algorithm, invented in the late 1940s, had fascinated optimization researchers for many years because its performance on practical problems is usually far better than the theoretical worst case. A new class of algorithms known as interior-point methods was the subject of intense theoretical and practical investigation during the period 1984 1995, with practical codes first appearing around 1989. These methods appeared to be faster than simplex on large problems, but the advent of a serious rival spurred significant improvements in simplex codes. Today, the relative merits of the two approaches on any given problem depend strongly on the particular geometric and algebraic properties of the problem. In general, however, good interior-point codes continue to perform as well or better than good simplex codes on larger problems when no prior information about the solution is available. When such  warm start information is available, however, as is often the case in solving continuous relaxations of integer linear programs in branch-and-bound algorithms, simplex methods are able to make much better use of it than interior-point methods. Further, a number of good interior-point codes are freely available for research purposes, while the few freely available simplex codes are not quite competitive with the best commercial codes. The simplex algorithm generates a sequence of feasible iterates  EMBED Equation.3 for the primal problem, where each iterate typically has the same number of nonzero (strictly positive) components as there are rows in  EMBED Equation.3 . We use this iterate to generate dual variables  EMBED Equation.3 and  EMBED Equation.3 such that two other optimality conditions are satisfied, namely,  EMBED Equation.3  If the remaining condition  EMBED Equation.3 is also satisfied, then the solution has been found and the algorithm terminates. Otherwise, we choose one of the negative components of  EMBED Equation.3 and allow the corresponding component of  EMBED Equation.3  to increase from zero. To maintain feasibility of the equality constraint  EMBED Equation.3  the components that were strictly positive in  EMBED Equation.3 will change. One of them will become zero when we increase the new component to a sufficiently large value. When this happens, we stop and denote the new iterate by  EMBED Equation.3 . Each iteration of the simplex method is relatively inexpensive. It maintains a factorization of the submatrix of  EMBED Equation.3  that corresponds to the strictly positive components of  EMBED Equation.3 (a square matrix  EMBED Equation.3 known as the basis(and updates this factorization at each step to account for the fact that one column of  EMBED Equation.3 has changed. Typically, simplex methods converge in a number of iterates that is about two to three times the number of columns in  EMBED Equation.3 . Interior-point methods proceed quite differently, applying a Newton-like algorithm to the three equalities in the optimality conditions and taking steps that maintain strict positivity of all the  EMBED Equation.3  and  EMBED Equation.3  components. It is the latter feature that gives rise to the term  interior-point (the iterates are strictly interior with respect to the inequality constraints. Each interior-point iteration is typically much more expensive than a simplex iteration, since it requires refactorization of a large matrix of the form  EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3 are diagonal matrices whose diagonal elements are the components of the current iterates  EMBED Equation.3 and  EMBED Equation.3 , respectively. The solutions to the primal and dual problems are generated simultaneously. Typically, interior-point iterates converge in between 10 and 100 iterations. Codes can differ in a number of important respects, apart from the different underlying algorithm. All practical codes include presolvers, which attempt to reduce the dimension of the problem by determining the values of some of the primal and dual variables without applying the algorithm. As a simplex example, suppose that the linear program contains the constraints  EMBED Equation.3  then the only possible values for the three variables are  EMBED Equation.3  These variables can be fixed and deleted from the problem, along with the three corresponding columns of  EMBED Equation.3 and the three components of  EMBED Equation.3 . Presolve techniques have become quite sophisticated over the years, though little has been written about them because of their commercial value. An exception is the paper of Andersen and Andersen (1995). For information on specific codes, refer to the online resources mentioned earlier; in particular, the NEOS Software Guide, the Linear Programming FAQ, and the benchmarks maintained by Hans Mittelmann. Modern, widely used commercial simplex codes include CPLEX and the XPRESS-MP. Both these codes accept input in the industry-standard MPS format, and also in their own proprietary formats. Both have interfaces to various modeling languages, and also a  callable library interface that allows users to set up, modify, and solve problems by means of function calls from C or FORTRAN code. Both packages are undergoing continual development. Freely available simplex codes are usually of lower quality, with the exception of SOPLEX. This is a C++ code, written as a thesis project by Roland Wunderling, that can be found at www.zib.de/Optimization/Software/Soplex/. The code MINOS is available to nonprofit and academic researchers for a nominal fee. Commercial interior-point solvers are available as options in the CPLEX and XPRESS-MP packages. However, a number of highly competitive codes are available free for research and noncommercial use, and can for the most part be obtained through the Web. Among these are BPMPD, PCx, COPLLP, LOQO, HOPDM, and LIPSOL. See Mittelmann s benchmark page for comparisons of these codes and links to their web sites. These codes mostly charge a license fee for commercial use, but it is typically lower than for fully commercial packages. All can read MPS files, and most are interfaced to modeling languages. LIPSOL is programmed in Matlab (with the exception of the linear equations solver), while the other codes are written in C and/or FORTRAN. A fine reference on linear programming, with an emphasis on the simplex method, is the book of Chvatal (1983). An online Java applet that demonstrates the operation of the simplex method on small user-defined problems can be found at www.mcs.anl.gov/otc/Guide/CaseStudies/simplex/. Wright (1997) gives a description of practical interior-point methods. Modeling Languages From the user s point of view, the efficiency of the algorithm or the quality of the programming may not be the critical factors in determining the usefulness of the code. Rather, the ease with which it can be interfaced to his particular applications may be more important; weeks of person-hours may be more costly to the enterprise than a few hours of time on a computer. The most suitable interface depends strongly on the particular application and on the context in which it is solved. For users that are well acquainted with a spreadsheet interface, for instance, or with MATLAB, a code that can accept input from these sources may be invaluable. For users with large legacy modeling codes that set up and solve optimization problems by means of subroutine calls, substitution of a more efficient package that uses more or less the same subroutine interface may be the best option. In some disciplines, (JP s biology/chemistry pointer) application-specific modeling languages allow problems to be posed in a thoroughly intuitive way. In other cases, application-specific graphical user interfaces may be more appropriate. For general optimization problems, a number of high-level modeling languages have become available that allow problems to be specified in intuitive terms, using data structures, naming schemes, and algebraic relational expressions that are dictated by the application and model rather than by the input requirements of the optimization code. Typically, a user starting from scratch will find the process of model building more straightforward and bug free with such a modeling language than, say, a process of writing FORTRAN code to pack the data into one-dimensional arrays, turning the algebraic relations between the variables into FORTRAN expressions involving elements of these arrays, and writing more code to interpret the output from the optimization routine in terms of the original application. The following simple example in AMPL demonstrates the usefulness of a modeling language (see Fourer, Gay, and Kernighan (1993), page 11). The application is to a steel production model, in which the aim is to maximize profit obtained from manufacturing a number of steel products by choosing the amount of each product to manufacture, subject to restrictions on the maximum demands for each product and the time available in each work week to manufacture them. The following file is an AMPL  model file that specifies the variables, the parameters that quantify aspects of the model, and the constraints and objective. set PROD; param rate {PROD} >0; param avail >= 0; param profit {PROD}; param market{PROD}; var Make {p in PROD} >= 0, <= market[p]; maximize total_profit: sum {p in PROD} profit[p] *Make[p]; subject to Time: sum {p in PROD} (1/rate[p]) * Make[p] <= avail; PROD is the collection of possible products that can be manufactured, while rate, profit and market are the rate at which each product can be manufactured, the profit on each product, and the maximum demand for each product, respectively. avail represents the total time available for manufacturing. Make is the variable in the problem, representing the amount of each product to be manufactured. In its definition, each element of Make is constrained to lie between zero and the maximum demand for the product in question. The last two lines of the model file specify the objective and constraint in a self-evident fashion. The actual values of the parameters can be assigned by means of additional statements in this file, or in a separate  data file. For instance, the following data file specifies parameters for two products, bands and coils: set PROD := bands coils; param: rate profit market := bands 200 25 6000 coils 140 30 4000; param avail := 40; These statements specify that the market[bands] is 6000, profit[bands] is 25, and so on. An interactive AMPL session would proceed by invoking commands to read these two files and then invoking an option solver command to choose the linear programming solver to be used (for example, CPLEX or PCx) together with settings for parameters such as stopping tolerances, etc, that the user may wish to change from their defaults. A solve command would then solve the problem (and report messages passed through from the underlying optimization code). Results can be inspected by invoking the display command. For the above example, the command display Make invoked after the problem has been solved would produce the following output: Make [*] := bands 6000 coils 0 ; Note from this example the intuitive nature of the algebraic relations, and the fact that we could index the parameter arrays by the indices bands and coils, rather than the numerical indices 1 and 2 that would be required if we were programming in FORTRAN. Note too that additional products can be added to the mix without changing the model file at all. Of course, the features of AMPL are much more extensive than the simple example above allows us to demonstrate. The web site  HYPERLINK http://www.ampl.com www.ampl.com contains a great deal of information about the language and the optimization software to which it is linked, and allows users to solve their own simple models online. Numerous other modeling languages and systems can be found on the online resources described above, particularly the NEOS Software Guide and the linear and nonlinear programming FAQ s. We mention in particular AIMMS (Bisschop and Entriken (1993)) which has a built in graphical interface; GAMS (www.gams.com), a well established system available with support for linear, nonlinear, and mixed-integer programming and newly added procedural features; and MPL, a Windows-based system whose web site  HYPERLINK http://www.maximal-usa.com www.maximal-usa.com contains a comprehensive tutorial and a free student version of the language. Other Input Formats The established input format for linear programming problems has from the earliest days been MPS, a column oriented format (well suited to 1950s card readers) in which names are assigned to each primal and dual variable, and the data elements that define the problem are assigned in turn. Test problems for linear programming are still distributed in this format. It has significant disadvantages, however. The format is non-intuitive and the files are difficult to modify. Moreover, it restricts the precision to which numerical values can be specified. The format survives only because no universally accepted standard has yet been developed to take its place. As mentioned previously, vendors such as CPLEX and XPRESS have their own input formats, which avoid the pitfalls of MPS. These formats lack the portability of the modeling languages described above, but they come bundled with the code, and may be attractive for users willing to make a commitment to a single vendor. For nonlinear programming, SIF (the standard input format) was proposed by the authors of the LANCELOT code in the early 1990s. SIF is somewhat hamstrung by the fact that it is compatible with MPS. SIF files have a similar look to MPS files, except that there are a variety of new keywords for defining variables, groups of variables, and the algebraic relationships between them. For developers of nonlinear programming software, SIF has the advantage that a large collection of test problems(the CUTE test set(is available in this format. For users, however, formulating a model in SIF is typically much more difficult than using one of the modeling languages of the previous section. For complete information about SIF, see www.numerical.rl.ac.uk/lancelot/sif/sifhtml.html Nonlinear Programming Nonlinear programming problems are constrained optimization problems with nonlinear objective and/or constraint functions. However, we still assume that all functions in question are smooth (typically, at least twice differentiable), and that the variables are all real numbers. If any of the variables are required to take on integer values, the problem is a (mixed-) integer nonlinear programming problem, a class that we will not consider in this paper. For purposes of description, we use the following formulation of the problem:  EMBED Equation.3 , where  EMBED Equation.3 is a vector of  EMBED Equation.3 real variables,  EMBED Equation.3 is a smooth real-valued function, and  EMBED Equation.3 and  EMBED Equation.3 are smooth functions with dimension  EMBED Equation.3 and  EMBED Equation.3 , respectively. Algorithms for nonlinear programming problems are more varied than those for linear programming. The major approaches represented in production software packages are sequential quadratic programming, reduced gradient, sequential linearly constrained, and augmented Lagrangian methods. (The latter is also known as the method of multipliers.) Extension of the successful interior-point approaches for linear programming to the nonlinear problem is the subject of intense ongoing investigation among optimization researchers, but little production software for these approaches is yet available. The use of nonlinear models may be essential in some applications, since a linear or quadratic model may be too simplistic and therefore produce useless results. However, there is a price to pay for using the more general nonlinear paradigm. For one thing, most algorithms cannot guarantee convergence to the global minimum, i.e., the value EMBED Equation.3 that minimizes EMBED Equation.3 over the entire feasible region. At best, they will find a point that yields the smallest value of  EMBED Equation.3 over all points in some feasible neighborhood of itself. (An exception occurs in convex programming, in which the functions  EMBED Equation.3 and  EMBED Equation.3 are convex, while  EMBED Equation.3 are linear. In this case, any local minimizer is also a global minimizer. Note that linear programming is a special case of convex programming.) The problem of finding the global minimizer, though an extremely important one in some applications such as molecular structure determination, is very difficult to solve. While several general algorithmic approaches for global optimization are available, they are invariably implemented in a way that exploits heavily the special properties of the underlying application, so that there is a fair chance that they will produce useful results in a reasonable amount of computing time. We refer to Floudas and Pardalos (1992) and the journal Global Optimization for information on recent advances in this area. A second disadvantage of nonlinear programming over linear programming is that general-purpose software is somewhat less effective because the nonlinear paradigm encompasses such a wide range of problems with a great number of potential pathologies and eccentricities. Even when we are close to a minimizer  EMBED Equation.3 , algorithms may encounter difficulties because the solution may be degenerate, in the sense that certain of the active constraints become dependent, or are only weakly active. Curvature in the objective or constraint functions (a second-order effect not present in linear programming), and differences in this curvature between different directions, can cause difficulties for the algorithms, especially when second derivative information is not supplied by the user or not exploited by the algorithm. Finally, some of the codes treat the derivative matrices as dense, which means that they the maximum dimension of the problems they can handle is somewhat limited. However, most of the leading codes, including LANCELOT, MINOS, SNOPT, and SPRNLP are able to exploit sparsity, and are therefore equipped to handle large-scale problems. Algorithms for special cases of the nonlinear programming problem, such as problems in which all constraints are linear or the only constraints are bounds on the components of  EMBED Equation.3 , tend to be more effective than algorithms for the general problem because they are more able to exploit the special properties. (We discuss a few such special cases below.) Even for problems in which the constraints are nonlinear, the problem may contain special structures that can be exploited by the algorithm or by the routines that perform linear algebra operations at each iteration. An example is the optimal control problem (arising, for example, in model predictive control), in which the equality constraint represents a nonlinear model of the  plant , and the inequalities represent bounds and other restrictions on the states and inputs. The Jacobian (matrix of first partial derivatives of the constraints) typically has a banded structure, while the Hessian of the objective is symmetric and banded. Linear algebra routines that exploit this bandedness, or dig even deeper and exploit the control origins of the problem, are much more effective than general routines on such problems. Local solutions of the nonlinear program can be characterized by a set of optimality conditions analogous to those described above for the linear programming problem. We introduce Lagrange multipliers  EMBED Equation.3  and  EMBED Equation.3 for the constraints  EMBED Equation.3  and  EMBED Equation.3 , respectively, and write the Lagrangian function for this problem as  EMBED Equation.3  The first-order optimality conditions (commonly known as the KKT conditions) are satisfied at a point  EMBED Equation.3  if there exist multiplier vectors  EMBED Equation.3  and  EMBED Equation.3  such that  EMBED Equation.3  The active constraints are those for which equality holds at  EMBED Equation.3 . All the components of  EMBED Equation.3  are active by definition, while the active components of  EMBED Equation.3  are those for which  EMBED Equation.3  When the constraint gradients satisfy certain regularity conditions at  EMBED Equation.3 , the KKT conditions are necessary for  EMBED Equation.3  to be a local minimizer of the nonlinear program, but not sufficient. A second-order sufficient condition is that the Hessian of the Lagrangian, the matrix  EMBED Equation.3 , has positive curvature along all directions that lie in the null space of the active constraint gradients, for some choice of multipliers  EMBED Equation.3  and  EMBED Equation.3  that satisfy the KKT conditions. That is, we require  EMBED Equation.3  for all vectors  EMBED Equation.3  such that  EMBED Equation.3  and  EMBED Equation.3  for all active indices  EMBED Equation.3 . The sequential quadratic programming (SQP) approach has been investigated extensively from a theoretical point of view and is the basis of several important practical codes, including NPSOL and the more recent SNOPT. It works by approximating the nonlinear programming problem by a quadratic program around the current iterate  EMBED Equation.3 , that is,  EMBED Equation.3  where EMBED Equation.3 is a symmetric matrix (usually positive definite) that contains exact or approximate second-order information about the objective and constraint functions. There are many modifications of this basic scheme. For instance, a trust-region bound limiting the length of the step  EMBED Equation.3 may be added to the model, or the linear constraints may be adjusted so that the current step is not required to remedy all the infeasibility in the current iterate  EMBED Equation.3 . The approximate Hessian  EMBED Equation.3  can be chosen in a number of ways. Local quadratic convergence can be proved under certain assumptions if this matrix is set to the Hessian of the Lagrangian, that is,  EMBED Equation.3 evaluated at the primal iterate EMBED Equation.3 and the current estimates  EMBED Equation.3  of the Lagrange multiplier vectors. The code SPRNLP allows users to select this value for  EMBED Equation.3 , provided that they are willing to supply the second derivative information. Alternatively,  EMBED Equation.3  can be a quasi-Newton approximation to the Lagrangian Hessian. Update strategies that yield local superlinear convergence are well known, and are implemented in dense codes such as NPSOL, DONLP2, NLPQL, and are available as an option in a version of SPRNLP that does not exploit sparsity. SNOPT also uses quasi-Newton Hessian approximations, but unlike the codes just mentioned it is able to exploit sparsity and is therefore better suited to large-scale problems. Another quasi-Newton variant is to maintain an approximation to the reduced Hessian, the two-sided projection of this matrix onto the null space of the active constraints. The latter approach is particularly efficient when the dimension of this null space is small in relation to the number of components of  EMBED Equation.3 , as is the case in many process control problems, for instance. The approach does not appear to be implemented in general-purpose SQP software, however. To ensure that the algorithm converges to a point satisfying the KKT conditions from any starting point, the basic SQP algorithm must be enhanced by the addition of a  global convergence strategy. Usually, this strategy involves a merit function, whose purposes is to evaluate the desirability of a given iterate  EMBED Equation.3  by accounting for its objective value and the amount by which it violates the constraints. The commonly used  EMBED Equation.3  penalty function simply forms a weighted average of the objective and the constraint violations, as follows:  EMBED Equation.3  where  EMBED Equation.3  is the vector of length  EMBED Equation.3  whose elements are  EMBED Equation.3 and  EMBED Equation.3  is a positive parameter. The simplest algorithm based on this function fixes  EMBED Equation.3  and insists that all steps produce a  sufficient decrease in the value of  EMBED Equation.3 . Line search or trust region strategies are applied to ensure that steps with the required property can be found whenever the current point  EMBED Equation.3  does not satisfy the KKT conditions. More sophisticated strategies contain mechanisms for adjusting the parameter  EMBED Equation.3  and for ensuring that the fast local convergence properties are not compromised by the global convergence strategy. We note that the terminology can be confusing( global convergence in this context refers to convergence to a KKT point from any starting point, and not to convergence to a global minimizer. For more information on SQP, we refer to the review paper of Boggs and Tolle (1996), and Chapter 18 of Nocedal and Wright (1999). A second algorithmic approach is known variously as the augmented Lagrangian method or the method of multipliers. Noting that the first KKT condition, namely,  EMBED Equation.3 , requires EMBED Equation.3 to be a stationary point of the Lagrangian function  EMBED Equation.3 , we modify this function to obtain an augmented function for which  EMBED Equation.3  is not just a stationary point but also a minimizer. When only equality constraints are present (that is,  EMBED Equation.3  is vacuous), the augmented Lagrangian function has the form  EMBED Equation.3  where  EMBED Equation.3  is a positive parameter. It is not difficult to show that if  EMBED Equation.3 is set to its optimal value  EMBED Equation.3  (the value that satisfies the KKT conditions) and  EMBED Equation.3  is sufficiently large, that  EMBED Equation.3  is a minimizer of  EMBED Equation.3 . Intuitively, the purpose of the squared-norm term is to add positive curvature to the function  EMBED Equation.3  in just those directions in which it is needed(the directions in the range space of the active constraint gradients. (We know already from the second-order sufficient conditions that the curvature of  EMBED Equation.3  in the null space of the active constraint gradients is positive.) In the augmented Lagrangian method, we exploit this property by alternating between steps of two types: Fixing  EMBED Equation.3  and  EMBED Equation.3 , and finding the value of  EMBED Equation.3  that approximately minimizes  EMBED Equation.3 ; Updating  EMBED Equation.3  to make it a better approximation to  EMBED Equation.3 . The update formula for  EMBED Equation.3  has the form  EMBED Equation.3  where  EMBED Equation.3  is the approximate minimizing value just calculated. Simple constraints such as bounds or linear equalities can be treated explicitly in the subproblem, rather than included in the second and third terms of  EMBED Equation.3 . (In LANCELOT, bounds on components of  EMBED Equation.3  are treated in this manner.) Practical augmented Lagrangian algorithms also contain mechanisms for adjusting the parameter EMBED Equation.3  and for replacing the squared norm term  EMBED Equation.3  by a weighted norm that more properly reflects the scaling of the constraints and their violations at the current point. When inequality constraints are present in the problem, the augmented Lagrangian takes on a slightly more complicated form that is nonetheless not difficult to motivate. We define the function  EMBED Equation.3 as follows:  EMBED Equation.3  The definition of  EMBED Equation.3  is then modified to incorporate the inequality constraints as follows:  EMBED Equation.3  The update formula for the approximate multipliers  EMBED Equation.3  is  EMBED Equation.3  See the references below for details on derivation of this form of the augmented Lagrangian. The definitive implementation of the augmented Lagrangian approach for general-purpose nonlinear programming problems is LANCELOT. It incorporates sparse linear algebra techniques, including preconditioned iterative linear solvers, making it suitable for large-scale problems. The subproblem of minimizing the augmented Lagrangian with respect to  EMBED Equation.3  is a bound-constrained minimization problem, which is solved by an enhanced gradient projection technique. Problems can be passed to Lancelot via subroutine calls, SIF input files, and AMPL. For theoretical background on the augmented Lagrangian approach, consult the books of Bertsekas (1982, 1995), and Conn, Gould, and Toint (1992), the authors of LANCELOT. The latter book is notable mainly for its pointers to the papers of the same three authors in which the theory of Lancelot is developed. A brief derivation of the theory appears in Chapter 17 of Nocedal and Wright (1999). (Note that the inequality constraints in this reference are assumed to have the form  EMBED Equation.3  rather than  EMBED Equation.3 , necessitating a number of sign changes in the analysis.) Interior-point solvers for nonlinear programming are the subjects of intense current investigation. An algorithm of this class, known as the sequential unconstrained minimization technique (SUMT) was actually proposed in the 1960s, in the book of Fiacco and McCormick (1968). The idea at that time was to define a barrier-penalty function for the NLP as follows:  EMBED Equation.3 where  EMBED Equation.3  is a small positive parameter. Given some value of  EMBED Equation.3 , the algorithm finds an approximation to the minimizer  EMBED Equation.3  of  EMBED Equation.3 . It then decreases  EMBED Equation.3  and repeats the minimization process. Under certain assumptions, one can show that  EMBED Equation.3  as  EMBED Equation.3  so the sequence of iterates generated by SUMT should approach the solution of the nonlinear program provided that  EMBED Equation.3  is decreased to zero. The difficulties with this approach are that all iterates must remain strictly feasible with respect to the inequality constraints (otherwise the log functions are not defined), and the subproblem of minimizing  EMBED Equation.3  becomes increasingly difficult to solve as  EMBED Equation.3  becomes small, as the Hessian of this function becomes highly ill conditioned and the radius of convergence becomes tiny. Many implementations of this approach were attempted, including some with enhancements such as extrapolation to obtain good starting points for each value of  EMBED Equation.3 . However, the approach does not survive in the present generation of software, except through its profound influence on the interior-point research of the past 15 years. Some algorithms for nonlinear programming that have been proposed in recent years contain echoes of the barrier function  EMBED Equation.3 , however. For instance, the NITRO algorithm (Byrd, Gilbert, and Nocedal (1996)) reformulates the subproblem for a given positive value of  EMBED Equation.3  as follows:  EMBED Equation.3  NITRO then applies a trust-region SQP algorithm for equality constrained optimization to this problem, choosing the trust region to have the form  EMBED Equation.3  where the diagonal matrix EMBED Equation.3  and the trust-region radius  EMBED Equation.3  are chosen so that the step  EMBED Equation.3  does not violate strict positivity of the  EMBED Equation.3  components, that is,  EMBED Equation.3  NITRO is available through the NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server/ www.mcs.anl.gov/neos/Server/ . The user is required to specify the problem by means of FORTRAN subroutines to evaluate the objective and constraints. Derivatives are obtained automatically by means of ADIFOR. An alternative interior-point approach is closer in spirit to the successful primal-dual class of linear programming algorithms. These methods generate iterates by applying Newton-like methods to the equalities in the KKT conditions. After introducing the slack variables  EMBED Equation.3  for the inequality constraints, we can restate the KKT conditions as follows:  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3 are diagonal matrices formed from the vectors  EMBED Equation.3  and  EMBED Equation.3 , respectively, while  EMBED Equation.3  is the vector  EMBED Equation.3 . We generate a sequence of iterates  EMBED Equation.3  satisfying the strict inequality  EMBED Equation.3  by applying a Newton-like method to the system of nonlinear equations formed by the first four conditions above. Modification of this basic approach to ensure global convergence is the major challenge associated with this class of solvers; the local convergence theory is relatively well understood. Merit functions can be used, along with line searches and modifications to the matrix in the equations that are solved for each step, to ensure that each step at least produces a decrease in the merit function. However, no fully satisfying complete theory has yet been proposed. The code LOQO implements a primal-dual approach for nonlinear programming problems. It requires the problem to be specified in AMPL, whose built-in automatic differentiation features are used to obtain the derivatives of the objective and constraints. LOQO is also available through the NEOS Server at  HYPERLINK http://www.mcs.anl.gov/neos/Server/ www.mcs.anl.gov/neos/Server/ , and or can be obtained for a variety of platforms. The reduced gradient approach has been implemented in several codes that have been available for some years, notably, CONOPT and LSGRG2. This approach uses the formulation in which only bounds and equality constraints are present. (Any nonlinear program can be transformed to this form by introducing slacks for the inequality constraints and constraining the slacks to be nonnegative.) Reduced gradient algorithms partition the components of  EMBED Equation.3  into three classes: basic, fixed, and superbasic. The equality constraint  EMBED Equation.3  is used to eliminate the basic components from the problem by expressing them implicitly in terms of the fixed and superbasic components. The fixed components are those that are fixed at one of their bounds for the current iteration. The superbasics are the components that are allowed to move in a direction that reduces the value of the objective  EMBED Equation.3 . Strategies for choosing this direction are derived from unconstrained optimization; they include steepest descent, nonlinear conjugate gradient, and quasi-Newton strategies. Both CONOPT and LSGRG2 use sparse linear algebra techniques during the elimination of the basic components, making them suitable for large-scale problems. While these codes have found use in many engineering applications, their performance is often slower than competing codes based on SQP and augmented Lagrangian algorithms. Finally, we mention MINOS, a code that has been available for many years in a succession of releases, and that has proved its worth in a great many engineering applications. When the constraints are linear, MINOS uses a reduced gradient algorithm, maintaining feasibility at all iterations and choosing the superbasic search direction with a quasi-Newton technique. When nonlinear constraints are present, MINOS forms linear approximations to them and replaces the objective with a projected augmented Lagrangian function in which the deviation from linearity is penalized. Convergence theory for this approach is not well established(the author admits that a reliable merit function is not known(but it appears to converge on most problems. The NEOS Guide page for SNOPT contains some guidance for users who are unsure whether to use MINOS or SNOPT. It describes problem features that are particularly suited to each of the two codes. Obtaining Derivatives One onerous requirement of some nonlinear programming codes has been their requirement that the user supply code for calculating derivatives of the objective and constraint functions. An important development of the past 10 years is that this requirement has largely disappeared. Modeling languages such as AMPL contain their own built-in systems for calculating first derivatives at specified values of the variable vector  EMBED Equation.3 , and supplying them to the underlying optimization code on request. Automatic differentiation software tools such as ADIFOR (Bischof et al. (1996)), which works with FORTRAN code, have been used to obtain derivatives from extremely complex  dusty deck function evaluation routines. In the NEOS Server, all of the nonlinear optimization routines (including LANCELOT, SNOPT, and NITRO) are linked to ADIFOR, so that the user needs only to supply FORTRAN code to evaluate the objective and constraint functions, not their derivatives. Other high quality software tools for automatic differentiation include ADOL-C (Griewank, Juedes, and Utke (1996)), ODYSSEE (Rostaing, Dalmas, and Galligo (1993)), and ADIC (Bischof, Roh, and Mauer (1997)). References Andersen, E. D. and Andersen, K. D. (1995). Presolving in linear programming. Math. Prog., 71, 221-245. Bertsekas, D. P. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York. Bertsekas, D. P. (1995). Nonlinear Programming. Athena Scientific. Bischof, C., Carle, A., Khademi, P., and Mauer, A. (1996). ADIFOR 2.0: Automatic differentiation of FORTRAN programs. IEEE Computational Science and Engineering, 3, 18-32. Bischof, C., Roh, L., and Mauer, A. (1997). ADIC: An extensible automatic differentiation tool for ANSI-C. Software-Practice and Experience, 27, 1427-1456. Bisschop, J. and Entriken, R. (1993). AIMMS: The Modeling System. Available from AIMMS web site at http://www.paragon.nl Boggs, P. T. and Tolle, J. W. (1996). Sequential quadratic programming, Acta Numerica, 4, 1-51. Byrd, R. H., Gilbert, J.-C., and Nocedal, J. (1996). A trust-region algorithm based on interior-point techniques for nonlinear programming. OTC Technical Report 98/06, Optimization Technology Center. (Revised, 1998.) Chvatal, V. (1983). Linear Programming. Freeman, New York. Conn, A. R., Gould, N. I. M., and Toint, Ph. L. (1992). LANCELOT: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A). Volume 17, Springer Series in Computational Mathematics, Springer-Verlag, New York. Czyzyk, J., Mesnier. M. P., and More , J. J. (1998). The NEOS Server. IEEE Journal on Computational Science and Engineering, 5, 68-75. Fiacco, A. V. and McCormick, G. P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Tecnhiques. John Wiley and Sons, New York. (Reprinted by SIAM Publications, 1990.) Floudas, C. and Pardalos, P., eds (1992). Recent Advances in Global Optimization. Princeton University Press, Princeton. Fourer, R., Gay, D. M., and Kernighan, B. W. (1993). AMPL: A Modeling Language for Mathematical Programming.The Scientific Press, San Francisco. Griewank, A., Juedes, D., and Utke, J. (1996). ADOL-C, A package for the automatic differentiation of algorithms written in C/C++. ACM Transactions on Mathematical Software, 22, 131-167. Luenberger, D. (1984). Introduction to Linear and Nonlinear Programming. Addison Wesley. Nash, S. and Sofer, A. (1996). Linear and Nonlinear Programming. McGraw-Hill. Nocedal, J. and Wright, S. J. (forthcoming,1999). Numerical Optimization. Springer, New York. Rostaing, N., Dalmas, S., and Galligo, A. (1993). Automatic differentiation in Odyssee. Tellus, 45a, 558-568. Wright, S. J. (1997). Primal-Dual Interior-Point Methods. 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52wɪ\3[K)7bfv`!wɪ\3[K)7b`!Hx5O ` ,N$n_é>@ - v$>C']&i $9r\`+NE !ڐ Y1l+JH{CLPL/nR~ ]yN1?|Z<[b1¸ k/ 4C#o2,./E7A[$~'Dd L 3670 745 mt 3673 748 L 3673 748 mt 3675 750 L 3675 750 mt 3678 752 L 3678 752 mt 3681 754 L 3681 754 mt 3683 756 L 3683 756 mt 3686 759 L 3686 759 mt 3688 760 L 3688 760 mt 3691 764 L 3691 764 mt 3692 765 L 3692 765 mt 3697 768 L 3697 768 mt 3697 769 L 3697 769 mt 3702 773 L 3702 773 mt 3702 773 L 3702 773 mt 3706 777 L 3706 777 mt 3707 778 L 3707 778 mt 3711 782 L 3711 782 mt 3713 783 L 3713 783 mt 3716 786 L 3716 786 mt 3718 788 L 3718 788 mt 3 FMicrosoft Equation 3.0 DS Equation Equation.39q9 799 L 3729 799 mt 3733 803 L 3733 803 mt 3734 804 L 3734 804 mt 3738 807 L 3738 807 mt 3740 809 L 3740 809 mt 3742 811 L 3742 811 mt 3745 814 L 3745 814 mt 3746 815 L 3746 815 mt 3750 819 L 3750 819 mt 3751 820 L 3751 820 mt 3755 824 L 3755 824 mt 3756 825 L 3756 825 mt 3759 828 L 3759 828 mt 3761 830 L 3761 830 mt 3763 832 L 3763 832 mt 3766 836 L 3766 836 mt 3767 837 L 3 837 mt 3772 841 L 3772 841 mt 3772 841 L 3772 841 mt 3776 845 L 3776 845 mt 3777 847 L 3777 847 mt 3780 849 L 3780 849 mt 3783 852 L 3783 852 mt 3784 854 L 3784 854 mt 3788 858 L 3788 858 mt 3788 858 L 3788 858 mt 3792 862 L 3792 862 mt 3793 864 L 3793 864 mt 3796 866 L 3796 866 mt 3799 870 L 3799 870 mt 3799 871 L 3799 871 mt 3803 875 L 3803 875 mt 3804 876 L 3804 876 mt 3807 879 L 3807 879 mt 3809 882 L 3809 882 mt 3811 883 L 3811 883 mt 3815 887 L 3815 887 mt II  92 mt 3820 894 L 3820 894 mt 3822 896 L 3822 896 mt 3826 900 L 3826 900 mt 3826 900 L 3826 900 mt 3830 904 L 3830 904 mt 3831 906 L 3831 906 mt 3833 909 L 3833 909 mt 3836 912 L 3836 912 mt 3837 913 L 3837 913 mt 3840 917 L 3840 917 mt 3842 919 L 3842 919 mt 3844 921 L 3844 921 mt 3847 925 L 3847 925 mt 3848 926 L 3848 926 mt 3851 930 L 3851 930 mt 3852 931 L 3852 931 mt 3855 934 L 3855 934 mt 3858 938 L 3858 938 mt 3858 938 L 42 L 3861 942 mt 3863 944 L 3863 944 mt 3865 947 L 3865 947 mt 3868 951 L 3868 951 mt 3868 951 L 3868 951 mt 3872 955 L 3872 955 mt 3874 958 L 3874 958 mt 3875 959 L 3875 959 mt 3878 964 L 3878 964 mt 3879 965 L 3879 965 mt 3882 968 L 3882 968 mt 3885 971 L 3885 971 mt 3885 972 L 3885 972 mt 3888 976 L 3888 976 mt 3890 978 L 3890 978 mt 3892 981 L 3892 981 mt 3895 985 L 3895 985 mt 3895 985 L 3895 985 mt 3898 989 L 3898 989 mt 3901 992 L 3901 992 mt FMicrosoft Equation 3.0 DS Equation Equation.39q911 1006 L 3911 1006 mt 3911 1007 L 3911 1007 mt 3914 1010 L 3914 1010 mt 3917 1014 L 3917 1014 mt 3917 1014 L 3917 1014 mt 3920 1019 L 3920 1019 mt 3922 1021 L 3922 1021 mt 3923 1023 L 3923 1023 mt 3926 1027 L 3926 1027 mt 3928 1029 L 3928 1029 mt 3929 1031 L 3929 1031 mt 3932 1036 L 3932 1036 mt 3933 1036 L 3933 1036 mt 3935 1040 L 3935 1040 mt 3938 1044 L 3938 1044 mt 3938 1044 L 3938 1044 mt 3941 1048 L1 1048 mt 3944 1052 L 3944 1052 mt 3944 1052 L 3944 1052 mt 3947 1057 L 3947 1057 mt 3949 1060 L 3949 1060 mt 3950 1061 L 3950 1061 mt 3953 1065 L 3953 1065 mt 3954 1067 L 3954 1067 mt 3956 1069 L 3956 1069 mt 3959 1074 L 3959 1074 mt 3960 1075 L 3960 1075 mt 3961 1078 L 3961 1078 mt 3964 1082 L 3964 1082 mt 3965 1083 L 3965 1083 mt 3967 1086 L 3967 1086 mt 3970 1091 L 3970 1091 mt 3970 1092 L 3970 1092 mt 3973 1095 L 3973 1095 mt 3975 1099 L 3975 1099 mt 3976 1100 L 3976 1100 mt 3978 1103 L 3978 1103 mII 1108 mt 3984 1112 L 3984 1112 mt 3986 1116 L 3986 1116 mt 3987 1116 L 3987 1116 mt 3989 1120 L 3989 1120 mt 3992 1124 L 3992 1124 mt 3992 1125 L 3992 1125 mt 3994 1129 L 3994 1129 mt 3997 1133 L 3997 1133 mt 3997 1133 L 3997 1133 mt 4000 1137 L 4000 1137 mt 4002 1141 L 4002 1141 mt 4003 1142 L 4003 1142 mt 4005 1146 L 4005 1146 mt 4007 1150 L 4007 1150 mt 4008 1151 L 4008 1151 mt 4010 1154 L 4010 1154 mt 4013 1158 L 4013 1158 mt 4013 1160 L 4013 1160 mt 4015 1163 B 8 S A5? 62eR}pv`!eR}p``!!x5O ` ,N"ЭS}  v$>.G1:u%29gȂƆZv];ڑIzCLTXLnRaˉWm]Ҙ~--qV 4f(s$S7bm{ZFbgʋw8h31"DdH@B 9 S A6? 1225 mt 4052 1226 L 4052 1226 mt 4054 1230 L 4054 1230 mt 4056 1234 L 4056 1234 mt 4056 1235 L 4056 1235 mt 4059 1239 L 4059 1239 mt 4061 1243 L 4061 1243 mt 4062 1245 L 4062 1245 mt 4063 1247 L 4063 1247 mt 4065 1251 L 4065 1251 mt 4067 1255 L 4067 1255 mt 4068 1256 L 4068 1256 mt 4070 1260 L 4070 1260 mt 4072 1264 L 4072 1264 mt 4073 1265 L 4073 1265 mt 4074 1268 L 4074 1268 mt 4077 1273 L 4077 1273 mt 4078 1275 L 4078 1275 mt 4079 1277 L 4079 1277 mt 4081 1281 L 4081 1281 mt 4083 1285 FMicrosoft Equation 3.0 DS Equation Equation.39q 4089 1296 mt 4090 1298 L 4090 1298 mt 4092 1302 L 4092 1302 mt 4094 1306 L 4094 1306 mt 4094 1307 L 4094 1307 mt 4096 1311 L 4096 1311 mt 4098 1315 L 4098 1315 mt 4099 1317 L 4099 1317 mt 4100 1319 L 4100 1319 mt 4102 1323 L 4102 1323 mt 4104 1328 L 4104 1328 mt 4105 1328 L 4105 1328 mt 4106 1332 L 4106 1332 mt 4108 1336 L 4108 1336 mt 4110 1340 L 4110 1340 mt 4110 1340 L 4110 1340 mt 4112 1345 L 4112 13454115 1349 L 4115 1349 mt 4115 1351 L 4115 1351 mt 4117 1353 L 4117 1353 mt 4118 1357 L 4118 1357 mt 4120 1361 L 4120 1361 mt 4121 1362 L 4121 1362 mt 4122 1366 L 4122 1366 mt 4124 1370 L 4124 1370 mt 4126 1374 L 4126 1374 mt 4126 1374 L 4126 1374 mt 4128 1378 L 4128 1378 mt 4130 1383 L 4130 1383 mt 4132 1386 L 4132 1386 mt 4132 1387 L 4132 1387 mt 4134 1391 L 4134 1391 mt 4136 1395 L 4136 1395 mt 4137 1398 L 4137 1398 mt 4138 1400 L 4138 1400 mt 4140 1404 L 4140 1404 mt 4142 1408 L 4142 1408 mt 4142 141$kIvI c(x)d"0mt 4147 1421 L 4147 1421 mt 4148 1422 L 4148 1422 mt 4149 1425 L 4149 1425 mt 4151 1429 L 4151 1429 mt 4153 1433 L 4153 1433 mt 4153 1435 L 4153 1435 mt 4154 1438 L 4154 1438 mt 4156 1442 L 4156 1442 mt 4158 1446 L 4158 1446 mt 4158 1447 L 4158 1447 mt 4160 1450 L 4160 1450 mt 4161 1455 L 4161 1455 mt 4163 1459 L 4163 1459 mt 4164 1460 L 4164 1460 mt 4165 1463 L 4165 1463 mt 4167 1467 L 4167 1467 mt 4168 1472 L 4168 1472 mt 4169 1473 L 4169 14772b쾸/&>v`!6쾸/&@" xcdd``~ @bD"L1JE `xX,56~) M @ Tv 7$# !L av@Hfnj_jBP~nbC3%@y mĕRl[] `5nAfܤ&`~@2_t1  b#ܤ 8d.hpC uk=pfdbR ,.IͅZf:: BBDd[@B : S A7? 896 1543 L 4196 1543 mt 4198 1548 L 4198 1548 mt 4199 1552 L 4199 1552 mt 4201 1556 L 4201 1556 mt 4201 1557 L 4201 1557 mt 4203 1560 L 4203 1560 mt 4204 1565 L 4204 1565 mt 4206 1569 L 4206 1569 mt 4207 1572 L 4207 1572 mt 4207 1573 L 4207 1573 mt 4209 1577 L 4209 1577 mt 4210 1582 L 4210 1582 mt 4212 1586 L 4212 1586 mt 4212 1587 L 4212 1587 mt 4213 1590 L 4213 1590 mt 4215 1594 L 4215 1594 mt 4216 1598 L 4216 1598 mt 4217 1603 L 4217 1603 mt 4217 1603 L 4217 1603 mt 4219 1607 L 4219 16 FMicrosoft Equation 3.0 DS Equation Equation.39q mt 4225 1624 L 4225 1624 mt 4226 1628 L 4226 1628 mt 4227 1632 L 4227 1632 mt 4228 1635 L 4228 1635 mt 4229 1637 L 4229 1637 mt 4230 1641 L 4230 1641 mt 4232 1645 L 4232 1645 mt 4233 1649 L 4233 1649 mt 4234 1651 L 4234 1651 mt 4234 1654 L 4234 1654 mt 4236 1658 L 4236 1658 mt 4237 1662 L 4237 1662 mt 4238 1666 L 4238 1666 mt 4239 1668 L 4239 1668 mt 4240 1670 L 4240 1670 mt 4241 1675 L 4241 1675 mt 4242 1L 4242 1679 mt 4244 1683 L 4244 1683 mt 4244 1686 L 4244 1686 mt 4245 1687 L 4245 1687 mt 4246 1692 L 4246 1692 mt 4247 1696 L 4247 1696 mt 4249 1700 L 4249 1700 mt 4250 1704 L 4250 1704 mt 4250 1704 L 4250 1704 mt 4251 1709 L 4251 1709 mt 4252 1713 L 4252 1713 mt 4254 1717 L 4254 1717 mt 4255 1721 L 4255 1721 mt 4255 1722 L 4255 1722 mt 4256 1725 L 4256 1725 mt 4257 1730 L 4257 1730 mt 4258 1734 L 4258 1734 mt 4260 1738 L 4260 1738 mt 4260 1741 L 4260 1741 mt 4261 1742 L 4261 1742 mt 4262 1747 L 4262 1$IԊI h(x)=0-59 L 4266 1759 mt 4266 1760 L 4266 1760 mt 4267 1764 L 4267 1764 mt 4268 1768 L 4268 1768 mt 4269 1772 L 4269 1772 mt 4270 1776 L 4270 1776 mt 4271 1780 L 4271 1780 mt 4271 1780 L 4271 1780 mt 4272 1785 L 4272 1785 mt 4273 1789 L 4273 1789 mt 4275 1793 L 4275 1793 mt 4276 1797 L 4276 1797 mt 4277 1801 L 4277 1801 mt 4277 1802 L 4277 1802 mt 4278 1806 L 4278 1806 mt 4279 1810 L 4279 1810 mt 4280 1814 L 4280 1814 mt 4281 1819 L 4281 1819 mt 4282 2c|ͱ yo]N?v`!7|ͱ yo]N`S xcdd``~ @bD"L1JE `xX,56~) M @ +j䆪aM,,He`7S?C&0] ZZ]<6J`Lc)6ӄA| [lA 27)? b.#ا7a)p!#ABLb4ηb3!5ס``ÙI)$5b6kKAwDd hB ; S A8? 924299 1895 mt 4300 1899 L 4300 1899 mt 4301 1903 L 4301 1903 mt 4302 1907 L 4302 1907 mt 4302 1912 L 4302 1912 mt 4303 1916 L 4303 1916 mt 4303 1916 L 4303 1916 mt 4304 1920 L 4304 1920 mt 4305 1924 L 4305 1924 mt 4306 1929 L 4306 1929 mt 4307 1933 L 4307 1933 mt 4308 1937 L 4308 1937 mt 4309 1941 L 4309 1941 mt 4309 1942 L 4309 1942 mt 4309 1946 L 4309 1946 mt 4310 1950 L 4310 1950 mt 4311 1954 L 4311 1954 mt 4312 1958 L 4312 1958 mt 4313 1962 L 4313 1962 mt 4314 1967 L 4314 1967 mt 4314 FMicrosoft Equation 3.0 DS Equation Equation.39q984 L 4317 1984 mt 4318 1988 L 4318 1988 mt 4318 1992 L 4318 1992 mt 4319 1996 L 4319 1996 mt 4320 1999 L 4320 1999 mt 4320 2001 L 4320 2001 mt 4321 2005 L 4321 2005 mt 4321 2009 L 4321 2009 mt 4322 2013 L 4322 2013 mt 4323 2017 L 4323 2017 mt 4323 2022 L 4323 2022 mt 4324 2026 L 4324 2026 mt 4325 2030 L 4325 2030 mt 4325 2030 L 4325 2030 mt 4326 2034 L 4326 2034 mt 4326 2039 L 4326 2039 mt 4327 2043 L 43273 mt 4328 2047 L 4328 2047 mt 4328 2051 L 4328 2051 mt 4329 2056 L 4329 2056 mt 4330 2060 L 4330 2060 mt 4330 2064 L 4330 2064 mt 4330 2064 L 4330 2064 mt 4331 2068 L 4331 2068 mt 4332 2073 L 4332 2073 mt 4332 2077 L 4332 2077 mt 4333 2081 L 4333 2081 mt 4334 2085 L 4334 2085 mt 4334 2089 L 4334 2089 mt 4335 2094 L 4335 2094 mt 4335 2098 L 4335 2098 mt 4336 2100 L 4336 2100 mt 4336 2102 L 4336 2102 mt 4337 2106 L 4337 2106 mt 4337 2111 L 4337 2111 mt 4338 2115 L 4338 2115 mt 4338 2119 L 4338 2119 mt 433vI8mI L(x,,)=f(x)+ T c(x)+ T h(x).4342 2149 L 4342 2149 mt 4343 2153 L 4343 2153 mt 4343 2157 L 4343 2157 mt 4344 2161 L 4344 2161 mt 4344 2166 L 4344 2166 mt 4345 2170 L 4345 2170 mt 4345 2174 L 4345 2174 mt 4346 2178 L 4346 2178 mt 4346 2183 L 4346 2183 mt 4346 2185 L 4346 2185 mt 4347 2187 L 4347 2187 mt 4347 2191 L 4347 2191 mt 4347 2195 L 4347 2195 mt 4348 2199 L 434XJ Se󎊞"v`!XJ Se󎊞" @L"|xAKA{fiA'`)56 \[= OA((XQ ; pIgL x yr%)BGGi gb,x஘]$ĨRRl7N>8G*Nvݺ[eGw젙do89|9c\ ]?zyLןKkW6Dd@B03 L 4356 2703 mt 4356 2707 L 4356 2707 mt 4356 2712 L 4356 2712 mt 4356 2716 L 4356 2716 mt 4355 2720 L 4355 2720 mt 4355 2724 L 4355 2724 mt 4355 2729 L 4355 2729 mt 4354 2733 L 4354 2733 mt 4354 2737 L 4354 2737 mt 4353 2741 L 4353 2741 mt 4353 2745 L 4353 2745 mt 4353 2750 L 4353 2750 mt 4352 2754 L 4352 2754 mt 4352 2758 L 4352 2758 mt 4352 2761 L 4352 2761 mt 4352 2762 L 4352 2762 mt 4351 2767 L 4351 2767 mt 4351 2771 L 4351 2771 mt 4350 2775 L 4350 2775 mt 4350 2779 L 4350 2779 mt FMicrosoft Equation 3.0 DS Equation Equation.39q348 2801 L 4348 2801 mt 4347 2805 L 4347 2805 mt 4347 2809 L 4347 2809 mt 4347 2813 L 4347 2813 mt 4346 2815 L 4346 2815 mt 4346 2817 L 4346 2817 mt 4346 2822 L 4346 2822 mt 4345 2826 L 4345 2826 mt 4345 2830 L 4345 2830 mt 4344 2834 L 4344 2834 mt 4344 2839 L 4344 2839 mt 4343 2843 L 4343 2843 mt 4343 2847 L 4343 2847 mt 4342 2851 L 4342 2851 mt 4342 2856 L 4342 2856 mt 4341 2860 L 4341 2860 mt 4341 2860 L1 2860 mt 4340 2864 L 4340 2864 mt 4340 2868 L 4340 2868 mt 4339 2872 L 4339 2872 mt 4339 2877 L 4339 2877 mt 4338 2881 L 4338 2881 mt 4338 2885 L 4338 2885 mt 4337 2889 L 4337 2889 mt 4337 2894 L 4337 2894 mt 4336 2898 L 4336 2898 mt 4336 2900 L 4336 2900 mt 4335 2902 L 4335 2902 mt 4335 2906 L 4335 2906 mt 4334 2911 L 4334 2911 mt 4334 2915 L 4334 2915 mt 4333 2919 L 4333 2919 mt 4332 2923 L 4332 2923 mt 4332 2927 L 4332 2927 mt 4331 2932 L 4331 2932 mt 4330 2936 L 4330 2936 mt 4330 2936 L 4330 2936 mIԊI  * 2949 L 4328 2949 mt 4328 2953 L 4328 2953 mt 4327 2957 L 4327 2957 mt 4326 2961 L 4326 2961 mt 4326 2966 L 4326 2966 mt 4325 2970 L 4325 2970 mt 4325 2970 L 4325 2970 mt 4324 2974 L 4324 2974 mt 4323 2978 L 4323 2978 mt 4323 2983 L 4323 2983 mt 4322 2987 L 4322 2987 mt 4321 2991 L 4321 2991 mt 4321 2995 L 4321 2995 mt 4320 2999 L 4320 2999 mt 4320 3001 L 4320 3001 mt 4319 3004 L 4319 3004 mt 4318 3008 L 4318 3008 mt 4318 3012 L 4318 3012 mt 4317 3016  = S A:? ;2GlK@D(ڟB#v`!lK@D(ڟBZ x=Pj@}3$AJO JB1'K~oۙItaٷ}0:(vB3Gφ*v7]{"r>Jw T&F$NKpYR0"Mgt~5IWdg)޺zs L S A;?3093 mt 4301 3097 L 4301 3097 mt 4300 3101 L 4300 3101 mt 4299 3105 L 4299 3105 mt 4298 3109 L 4298 3109 mt 4298 3109 L 4298 3109 mt 4297 3114 L 4297 3114 mt 4296 3118 L 4296 3118 mt 4295 3122 L 4295 3122 mt 4294 3126 L 4294 3126 mt 4293 3131 L 4293 3131 mt 4293 3133 L 4293 3133 mt 4292 3135 L 4292 3135 mt 4291 3139 L 4291 3139 mt 4290 3143 L 4290 3143 mt 4289 3148 L 4289 3148 mt 4288 3152 L 4288 3152 mt 4287 3156 L 4287 3156 mt 4287 3156 L 4287 3156 mt 4286 3160 L 4286 3160 mt 4285 3164 FMicrosoft Equation 3.0 DS Equation Equation.39q & MathTypePTimes New Roman-2 ,2 02 , 2 :maxSymbol-2 2 =2 +Times New Roman-2 s2 (c2 s2 A2 4 to 2 subjec2 .b Times New Roman-2 T2 TSymbol-2 l2 l & "Systemn-II  *I~~|Nx~x~x GDp f pw9>`xp`wDFLrh' M_EEMu Pwxpp wwpDDpwpppwpyx{pnvy8{{pG @D xn 39}gr}| M0_^[]USVW}wxwwwp~~~~pwwp www;wwDKpwwpywxpxx~x D `xww؅rFL@ @ <2LѓJCK>7+j(v`! ѓJCK>7+jZ xcdd``^ @bD"L1JE `x0 Yjl R A@v ^Դjx|K2B* R .ͤ `W01d++&10D(03kf#c@+ss >׿Q ..)I)$5N @dd`3tDd@B ? S A<? =srH} wxwpwwwwwwp p p; 3wxpwwpwx"2 wpw9onwr9~Lr}sE}|}tF@~$uF Uwxpwwwwwwwppwppwxpwwwx2""" w爈w~~~wpwwvwPTu U|XMU9uE,wvfffffpwwwwpwppwwwwwwwwwwwpwwwwpwwpw爈ww""wx~~xwwwwxwwwwpw FMicrosoft Equation 3.0 DS Equation Equation.39qwwwwwwwwpwpwwwwwwwwx~~xwwwwwx3wwwwwwwwwwwwpppwww؈EBEk0(G`E(ǀEwwfffff`wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwxwwwwwwwwwxwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwwwwww9996 999هMهMهMإl]}Ewwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwwwwwwwwwxwwww0wwwwwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwww9wwwwppL@wwwwwwwwwwwwwwwwwwwwwwwwpDDDffpwwwwwwwwFwwwwwwwxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwwwwpwwwwwwwwwwwwwwwwwwwwwwwwwwpt\@ppp@YQnfwwwwwwwwpwwwwphwwpwwwwwwwwwxpwwwwwpwwwwwwwwwwwwL$vIpI " x L(x * , * , * )=0,h(x * )=0,c(x * )d"0, * e"0, * () T c(x * )=0.pwwwwpwwwwwwwwnhha;f`w" tw~w~`~w~~`pp @vp pwwdGdfdDwwx2}_Ҹn|2H0ȡYv`!Q_Ҹn|2H0ȡ` .pxAkA{;MV\ xz'oV9Y@nZhA I"xܓ7&"Ѓ^DDo]glƙD0& ;y潙 D 8 9W:Swr5~8A"y g}9j1@_nY-}bXX^kv7<CͮmOP~iP.`2r_-}5p{tу捬UؾYt3БԊ^JT71띆әkO^ڿ #Sѧ"m}*~8T_8U[8\w]׊IO~"Wr*ٟ]>O6F\] \kv;kn)~nxĮg6r73'يpe WMfѶ9xSqk&۞m/2=Ñ%dpw=M)sl0ok{Vᗏ!)V"ёv;ӵ5y WDd@B @ S A9? >2:SWsFƨv`!SWsFƨdH xPQ=3ދH$(@PlBHln?|FI 5wvidrgsr%{2R]ntClM3%"y%ߩ*]+[Aƒ|* *UL0%bv@O3jxtDf`wwxxwww  wwwwwp;wwww;LĻpwwxww{g~w`w""'wsgxwwpnw`zwpwh`pYUpxww~wwwxwwwwwx~~~~~~~www  pwwwwwww;LĻpwwpwpwwn`x""ww0wwww`www`pUQn`g`@wwxxwww ppwwwp;3www;DDKpwwpwpwwxw{n~`y"" ~x FMicrosoft Equation 3.0 DS Equation Equation.39qww38ww;pwwwwpwwywww~w`r!"~vww`~p`pwwwpwwg`xAdhwwwwxwwwxwwwwwww w w wwp;33ww;pwwwpwwwxgww`ww! ~xx`~w`pwxwpp`@dwwwxwwwwwwwwwwpw w wpw 338w33333333ppwwwwxwww~~~`pw""9 " w~w~~~`~w~~~`pppwwp~fgFxwwwxwwwwwwwwwpp wpwp;wp33wppwwwwwwxwf~w`ps7r""""" wwwwwwwwwwwwwwwwwwwwwppwxwwwDhh`wwv~~~~~~~xwwwwwwwpwwp ppww ppwwpwwwwwpwwwwwwwwwww`ww#!"""" wwwwwwwwwpwwwwd@fwwvwwwwwwwwwwppwwwpkIvI x *wwwwwwp~w~wwwwwwwppwxppwwwww@' wwwv`wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwpwwwwwwwwwwwwwxw~xwwwwwwwws"""wwwwwwwwwwwwwwwwwwwwwwpwxxxwwwww@wwv~~~~~~~~`wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwpwwwwwwwwwwwps2"wwwwpwwwwwwwwpwwwwwwwpwpxwwwwwwwAwwwffffffff`wwwwwwwwwwwwwwwwww'R w 7wdW:{^D,(DQ )iI:Bhtܜ?f]Ś>W6DdB A S A=? ?2'DͿt%*v`!DͿt%*@Hx5O ` , NMQG'_@t(ԡ*(H7'7)|wAHr_r!sh@_Qe Be!=j2`ʧ}1եjh!SR=Yg"E@1#n(0:4:m7uҕ䡷 #zL'0e斘 L 3822 4104 mt 3820 4106 L 3820 4106 mt 3818 4108 L 3818 4108 mt 3815 4112 L 3815 4112 mt 3815 4113 L 3815 4113 mt 3811 4117 L 3811 4117 mt 3809 4118 L 3809 4118 mt 3807 4121 L 3807 4121 mt 3804 4124 L 3804 4124 mt 3803 4125 L 3803 4125 mt 3799 4129 L 3799 4129 mt 3799 4130 L 3799 4130 mt 3796 4134 L 3796 4134 mt 3793 4136 L 3793 4136 mt 3792 4138 L 3792 4138 mt 3788 4142 L 3788 4142 mt 3788 4142 L 3788 4142 mt 3784 4146 L 3784 4146 mt 3783 4148 L 3783 4148 mt 3780 4151 L 3780 4151 mt 3 FMicrosoft Equation 3.0 DS Equation Equation.39q7 4163 L 3767 4163 mt 3766 4164 L 3766 4164 mt 3763 4168 L 3763 4168 mt 3761 4170 L 3761 4170 mt 3759 4172 L 3759 4172 mt 3756 4175 L 3756 4175 mt 3755 4176 L 3755 4176 mt 3751 4180 L 3751 4180 mt 3750 4181 L 3750 4181 mt 3746 4185 L 3746 4185 mt 3745 4186 L 3745 4186 mt 3742 4189 L 3742 4189 mt 3740 4191 L 3740 4191 mt 3738 4193 L 3738 4193 mt 3734 4196 L 3734 4196 mt 3733 4197 L 3733 4197 mt 3729 4201 L 34201 mt 3729 4202 L 3729 4202 mt 3725 4206 L 3725 4206 mt 3723 4207 L 3723 4207 mt 3720 4210 L 3720 4210 mt 3718 4212 L 3718 4212 mt 3716 4214 L 3716 4214 mt 3713 4217 L 3713 4217 mt 3711 4218 L 3711 4218 mt 3707 4222 L 3707 4222 mt 3706 4223 L 3706 4223 mt 3702 4227 L 3702 4227 mt 3702 4227 L 3702 4227 mt 3697 4231 L 3697 4231 mt 3697 4232 L 3697 4232 mt 3692 4235 L 3692 4235 mt 3691 4236 L 3691 4236 mt 3688 4240 L 3688 4240 mt 3686 4241 L 3686 4241 mt 3683 4244 L 3683 4244 mt 3681 4246 L 3681 4246 mt PII h50 mt 3673 4252 L 3673 4252 mt 3670 4255 L 3670 4255 mt 3668 4256 L 3668 4256 mt 3664 4260 L 3664 4260 mt 3663 4261 L 3663 4261 mt 3659 4264 L 3659 4264 mt 3658 4265 L 3658 4265 mt 3654 4269 L 3654 4269 mt 3653 4269 L 3653 4269 mt 3648 4273 L 3648 4273 mt 3648 4273 L 3648 4273 mt 3643 4277 L 3643 4277 mt 3643 4278 L 3643 4278 mt 3638 4282 L 3638 4282 mt 3637 4282 L 3637 4282 mt 3632 4286 L 3632 4286 mt 3632 4286 L 3632 4286 mt 3627 4290 L 3627 4290 mt 3627 4290 L tߝ]٥%R*iDdB K S AD? @2&/KYIc=z~Xv`!/KYIc=z~ `Ƚ!x5O A |ݝ"buX)*xȂ bge'?Z)~BH2;L>]0p(ᳫ %R82EzTdC몼RuJ59:g9q_͏; O5+ F{$7+4.sM:TvwuK1wgWv/&Dd8|34 mt 3563 4337 L 3563 4337 mt 3562 4338 L 3562 4338 mt 3557 4341 L 3557 4341 mt 3557 4341 L 3557 4341 mt 3552 4345 L 3552 4345 mt 3551 4345 L 3551 4345 mt 3546 4348 L 3546 4348 mt 3545 4350 L 3545 4350 mt 3541 4352 L 3541 4352 mt 3538 4354 L 3538 4354 mt 3536 4356 L 3536 4356 mt 3532 4358 L 3532 4358 mt 3530 4359 L 3530 4359 mt 3525 4362 L 3525 4362 mt 3525 4362 L 3525 4362 mt 3519 4366 L 3519 4366 mt 3518 4367 L 3518 4367 mt 3514 4369 L 3514 4369 mt 3512 4371 L 3512 4371 mt 3509 4373 L FMicrosoft Equation 3.0 DS Equation Equation.39q498 4379 mt 3493 4382 L 3493 4382 mt 3491 4383 L 3491 4383 mt 3487 4385 L 3487 4385 mt 3483 4388 L 3483 4388 mt 3482 4389 L 3482 4389 mt 3476 4392 L 3476 4392 mt 3476 4392 L 3476 4392 mt 3471 4395 L 3471 4395 mt 3469 4396 L 3469 4396 mt 3466 4398 L 3466 4398 mt 3461 4400 L 3461 4400 mt 3460 4401 L 3460 4401 mt 3455 4404 L 3455 4404 mt 3453 4405 L 3453 4405 mt 3450 4407 L 3450 4407 mt 3446 4409 L 3446 4409 m44 4410 L 3444 4410 mt 3439 4412 L 3439 4412 mt 3438 4413 L 3438 4413 mt 3434 4415 L 3434 4415 mt 3430 4417 L 3430 4417 mt 3428 4418 L 3428 4418 mt 3423 4421 L 3423 4421 mt 3421 4422 L 3421 4422 mt 3417 4423 L 3417 4423 mt 3413 4426 L 3413 4426 mt 3412 4426 L 3412 4426 mt 3407 4429 L 3407 4429 mt 3404 4430 L 3404 4430 mt 3401 4431 L 3401 4431 mt 3396 4434 L 3396 4434 mt 3396 4434 L 3396 4434 mt 3391 4437 L 3391 4437 mt 3387 4438 L 3387 4438 mt 3385 4439 L 3385 4439 mt 3380 4442 L 3380 4442 mt 3377 4443 IdI c 4447 L 3369 4447 mt 3368 4447 L 3368 4447 mt 3364 4449 L 3364 4449 mt 3359 4451 L 3359 4451 mt 3358 4451 L 3358 4451 mt 3353 4454 L 3353 4454 mt 3349 4455 L 3349 4455 mt 3348 4456 L 3348 4456 mt 3342 4458 L 3342 4458 mt 3339 4460 L 3339 4460 mt 3337 4460 L 3337 4460 mt 3331 4463 L 3331 4463 mt 3329 4464 L 3329 4464 mt 3326 4465 L 3326 4465 mt 3321 4467 L 3321 4467 mt 3318 4468 L 3318 4468 mt 3315 4469 L 3315 4469 mt 3310 4471 L 3310 4471 mt 3307 4472 L 3307 4472 4473 mt 3299 4475 L 3299 4475 mt 3296 4477 L 3296 4477 mt 3294 4477 L 3294 4477 mt 3289 4479 L 3289 4479 mt 3285 4481 L 3285 4481 mt 3283 4481 L 3283 4481 mt 3278 4483 L 3278 4483 mt 3273 4485 L 3273 4485 mt 3272 4485 L 3272 4485 mt 3267 4487 L 3267 4487 mt 3262 4489 L 3262 4489 mt 3261 4489 L 3261 4489 mt 3256 4491 L 3256 4491 mt 3251 4493 L 3251 4493 mt 3249 4493 L 3249 4493 mt 3246 4495 L 3246 4495 mt 3240 4496 L 3240 4496 mt 3236 4498 L 3236 4498 mt 3235 4498 L 3235 4498 mt 3229 4500 FMicrosoft Equation 3.0 DS Equation Equation.39q 3213 4505 mt 3209 4506 L 3209 4506 mt 3208 4507 L 3208 4507 mt 3203 4508 L 3203 4508 mt 3197 4510 L 3197 4510 mt 3195 4510 L 3195 4510 mt 3192 4511 L 3192 4511 mt 3187 4513 L 3187 4513 mt 3181 4514 L 3181 4514 mt 3180 4515 L 3180 4515 mt 3176 4516 L 3176 4516 mt 3170 4517 L 3170 4517 mt 3165 4519 L 3165 4519 mt 3165 4519 L 3165 4519 mt 3160 4520 L 3160 4520 mt 3154 4522 L 3154 4522 mt 3149 4523 L 3149 45233149 4523 L 3149 4523 mt 3144 4524 L 3144 4524 mt 3138 4526 L 3138 4526 mt 3133 4527 L 3133 4527 mt 3132 4527 L 3132 4527 mt 3127 4529 L 3127 4529 mt 3122 4530 L 3122 4530 mt 3117 4531 L 3117 4531 mt 3115 4532 L 3115 4532 mt 3111 4532 L 3111 4532 mt 3106 4534 L 3106 4534 mt 3101 4535 L 3101 4535 mt 3096 4536 L 3096 4536 mt 3095 4536 L 3095 4536 mt 3090 4537 L 3090 4537 mt 3084 4538 L 3084 4538 mt 3079 4540 L 3079 4540 mt 3077 4540 L 3077 4540 mt 3074 4541 L 3074 4541 mt 3068 4542 L 3068 4542 mt 3063 454DII c i (x * )=0.E 4546 L 3047 4546 mt 3042 4547 L 3042 4547 mt 3036 4548 L 3036 4548 mt 3035 4549 L 3035 4549 mt 3031 4549 L 3031 4549 mt 3025 4550 L 3025 4550 mt 3020 4551 L 3020 4551 mt 3015 4552 L 3015 4552 mt 3012 4553 L 3012 4553 mt 3009 4553 L 3009 4553 mt 3004 4554 L 3004 4554 mt 2999 4555 L 2999 4555 mt 2993 4556 L 2993 4556 mt 2988 4557 L 2988 4557 mt 2987 4557 L 2987 4557 mt 2982 4558 L 2982 4558 mt 2977 4559 L 2977 455B L S AE? A2I|$@-cv`![I|$@-x`v 0)xRJ@fF (QQ>lBk X0VHQs#}^Lij}H\Dd75 4573 L 2875 4573 mt 2870 4574 L 2870 4574 mt 2867 4574 L 2867 4574 mt 2864 4574 L 2864 4574 mt 2859 4575 L 2859 4575 mt 2854 4575 L 2854 4575 mt 2848 4576 L 2848 4576 mt 2843 4577 L 2843 4577 mt 2837 4577 L 2837 4577 mt 2832 4578 L 2832 4578 mt 2829 4578 L 2829 4578 mt 2827 4578 L 2827 4578 mt 2821 4579 L 2821 4579 mt 2816 4579 L 2816 4579 mt 2811 4580 L 2811 4580 mt 2805 4580 L 2805 4580 mt 2800 4581 L 2800 4581 mt 2795 4582 L 2795 4582 mt 2789 4582 L 2789 4582 mt 2786 4582 L 2786 45 FMicrosoft Equation 3.0 DS Equation Equation.39q mt 2762 4584 L 2762 4584 mt 2757 4585 L 2757 4585 mt 2752 4585 L 2752 4585 mt 2746 4586 L 2746 4586 mt 2741 4586 L 2741 4586 mt 2737 4587 L 2737 4587 mt 2735 4587 L 2735 4587 mt 2730 4587 L 2730 4587 mt 2725 4588 L 2725 4588 mt 2719 4588 L 2719 4588 mt 2714 4588 L 2714 4588 mt 2709 4589 L 2709 4589 mt 2703 4589 L 2703 4589 mt 2698 4590 L 2698 4590 mt 2692 4590 L 2692 4590 mt 2687 4590 L 2687 4590 mt 2682 4L 2682 4591 mt 2680 4591 L 2680 4591 mt 2676 4591 L 2676 4591 mt 2671 4591 L 2671 4591 mt 2666 4592 L 2666 4592 mt 2660 4592 L 2660 4592 mt 2655 4592 L 2655 4592 mt 2650 4593 L 2650 4593 mt 2644 4593 L 2644 4593 mt 2639 4593 L 2639 4593 mt 2633 4594 L 2633 4594 mt 2628 4594 L 2628 4594 mt 2623 4594 L 2623 4594 mt 2617 4595 L 2617 4595 mt 2612 4595 L 2612 4595 mt 2611 4595 L 2611 4595 mt 2607 4595 L 2607 4595 mt 2601 4596 L 2601 4596 mt 2596 4596 L 2596 4596 mt 2590 4596 L 2590 4596 mt 2585 4596 L 2585 4IԊI x *I 2569 4597 L 2569 4597 mt 2564 4597 L 2564 4597 mt 2558 4598 L 2558 4598 mt 2553 4598 L 2553 4598 mt 2548 4598 L 2548 4598 mt 2542 4598 L 2542 4598 mt 2537 4599 L 2537 4599 mt 2531 4599 L 2531 4599 mt 2526 4599 L 2526 4599 mt 2525 4599 L 2525 4599 mt 2521 4599 L 2521 4599 mt 2515 4600 L 2515 4600 mt 2510 4600 L 2510 4600 mt 2505 4600 L 2505 4600 mt 2499 4600 L 2499 4600 mt 2494 4600 L 2494 4600 mt 2488 4601 L 2488 4601 mt 2483 4601 L 2483 4601 mt 2478 @B M S AF? B2:C9yT6a2V1v`!C9yT6a2VdH xPQ=3ދH$(@PlBHln?A ')t٥uɝ{9PRHvi% 1F7]ϔ|tl K+]T$T10a=ubϨI]02yu0_'#yev G4t!,g'qh I^"Msssz؆~ukRk6Dd@2370 4605 mt 2365 4605 L 2365 4605 mt 2360 4605 L 2360 4605 mt 2354 4605 L 2354 4605 mt 2349 4605 L 2349 4605 mt 2343 4605 L 2343 4605 mt 2338 4605 L 2338 4605 mt 2333 4605 L 2333 4605 mt 2327 4606 L 2327 4606 mt 2322 4606 L 2322 4606 mt 2317 4606 L 2317 4606 mt 2311 4606 L 2311 4606 mt 2306 4606 L 2306 4606 mt 2301 4606 L 2301 4606 mt 2295 4606 L 2295 4606 mt 2290 4607 L 2290 4607 mt 2284 4607 L 2284 4607 mt 2279 4607 L 2279 4607 mt 2274 4607 L 2274 4607 mt 2268 4607 L 2268 4607 mt 2263 FMicrosoft Equation 3.0 DS Equation Equation.39q608 L 2241 4608 mt 2238 4608 L 2238 4608 mt 2236 4608 L 2236 4608 mt 2231 4608 L 2231 4608 mt 2225 4608 L 2225 4608 mt 2220 4608 L 2220 4608 mt 2215 4608 L 2215 4608 mt 2209 4608 L 2209 4608 mt 2204 4608 L 2204 4608 mt 2198 4608 L 2198 4608 mt 2193 4608 L 2193 4608 mt 2188 4608 L 2188 4608 mt 2182 4609 L 2182 4609 mt 2177 4609 L 2177 4609 mt 2172 4609 L 2172 4609 mt 2166 4609 L 2166 4609 mt 2161 4609 L 21619 mt 2156 4609 L 2156 4609 mt 2150 4609 L 2150 4609 mt 2145 4609 L 2145 4609 mt 2139 4609 L 2139 4609 mt 2134 4609 L 2134 4609 mt 2129 4609 L 2129 4609 mt 2123 4609 L 2123 4609 mt 2118 4609 L 2118 4609 mt 2113 4609 L 2113 4609 mt 2107 4610 L 2107 4610 mt 2102 4610 L 2102 4610 mt 2096 4610 L 2096 4610 mt 2091 4610 L 2091 4610 mt 2086 4610 L 2086 4610 mt 2080 4610 L 2080 4610 mt 2075 4610 L 2075 4610 mt 2070 4610 L 2070 4610 mt 2064 4610 L 2064 4610 mt 2059 4610 L 2059 4610 mt 2053 4610 L 2053 4610 mt 204IԊI x *I L 2037 4610 mt 2032 4610 L 2032 4610 mt 2027 4611 L 2027 4611 mt 2021 4611 L 2021 4611 mt 2016 4611 L 2016 4611 mt 2011 4611 L 2011 4611 mt 2005 4611 L 2005 4611 mt 2000 4611 L 2000 4611 mt 1994 4611 L 1994 4611 mt 1989 4611 L 1989 4611 mt 1984 4611 L 1984 4611 mt 1978 4611 L 1978 4611 mt 1973 4611 L 1973 4611 mt 1968 4611 L 1968 4611 mt 1962 4611 L 1962 4611 mt 1957 4611 L 1957 4611 mt 1951 4611 L 1951 4611 mt 1946 4611 L 1946 4611 mt 1941 4612 L 194B N S AF? C2:C9yT6a2Vv`!C9yT6a2VdH xPQ=3ދH$(@PlBHln?A ')t٥uɝ{9PRHvi% 1F7]ϔ|tl K+]T$T10a=ubϨI]02yu0_'#yev G4t!,g'qh I^"Msssz؆~ukRk6;Dd@|B O S AG?2762 388 L 2762 388 mt 2768 389 L 2768 389 mt 2773 389 L 2773 389 mt 2778 389 L 2778 389 mt 2784 389 L 2784 389 mt 2789 389 L 2789 389 mt 2795 389 L 2795 389 mt 2800 389 L 2800 389 mt 2805 389 L 2805 389 mt 2811 389 L 2811 389 mt 2816 389 L 2816 389 mt 2821 389 L 2821 389 mt 2827 390 L 2827 390 mt 2832 390 L 2832 390 mt 2837 390 L 2837 390 mt 2843 390 L 2843 390 mt 2848 390 L 2848 390 mt 2854 390 L 2854 390 mt 2859 390 L 2859 390 mt 2864 390 L 2864  FMicrosoft Equation 3.0 DS Equation Equation.39q91 mt 2891 391 L 2891 391 mt 2897 391 L 2897 391 mt 2902 391 L 2902 391 mt 2907 391 L 2907 391 mt 2913 391 L 2913 391 mt 2918 391 L 2918 391 mt 2923 391 L 2923 391 mt 2929 391 L 2929 391 mt 2934 392 L 2934 392 mt 2940 392 L 2940 392 mt 2945 392 L 2945 392 mt 2950 392 L 2950 392 mt 2956 392 L 2956 392 mt 2961 392 L 2961 392 mt 2966 392 L 2966 392 mt 2971 392 L 2971 392 mt 29722 L 2972 392 mt 2977 392 L 2977 392 mt 2982 393 L 2982 393 mt 2988 393 L 2988 393 mt 2993 393 L 2993 393 mt 2999 393 L 2999 393 mt 3004 393 L 3004 393 mt 3009 393 L 3009 393 mt 3015 394 L 3015 394 mt 3020 394 L 3020 394 mt 3025 394 L 3025 394 mt 3031 394 L 3031 394 mt 3036 394 L 3036 394 mt 3042 395 L 3042 395 mt 3047 395 L 3047 395 mt 3052 395 L 3052 395 mt 3058 395 L 3058 395 mt 3063 395 L 3063 395 mt 3068 395 L 3068 395 mt 3074 396 L 3074 396 mt 3079 396 L 3079lkIvI " xx L(x * , * , * )06 397 L 3106 397 mt 3111 397 L 3111 397 mt 3117 397 L 3117 397 mt 3122 397 L 3122 397 mt 3127 398 L 3127 398 mt 3133 398 L 3133 398 mt 3138 398 L 3138 398 mt 3144 398 L 3144 398 mt 3149 399 L 3149 399 mt 3154 399 L 3154 399 mt 3160 399 L 3160 399 mt 3165 399 L 3165 399 mt 3170 400 L 3170 400 mt 3176 400 L 3176 400 mt 3181 400 L 3181 400 mt 318 D2 R<v`!y R< `p0Gxcdd``.ad``beV dX,XĐ )KRcgb u@P5< %! `fjvF+B2sSRsA.si# Śp2;盲A| 6aҤ@h~9Цܤ& 3]:8?'9aH0A Q0!d2rAI X2=ҋ \TNIȁ;+KRs!nAv3XRkLDd@L 3289 407 mt 3294 408 L 3294 408 mt 3299 408 L 3299 408 mt 3305 408 L 3305 408 mt 3310 409 L 3310 409 mt 3315 409 L 3315 409 mt 3315 409 L 3315 409 mt 3321 410 L 3321 410 mt 3326 410 L 3326 410 mt 3331 411 L 3331 411 mt 3337 411 L 3337 411 mt 3342 412 L 3342 412 mt 3348 412 L 3348 412 mt 3353 413 L 3353 413 mt 3358 413 L 3358 413 mt 3361 413 L 3361 413 mt 3364 414 L 3364 414 mt 3369 414 L 3369 414 mt 3374 415 L 3374 415 mt 3380 415 L 3380 415 mt 33 FMicrosoft Equation 3.0 DS Equation Equation.39q 418 L 3402 418 mt 3407 418 L 3407 418 mt 3412 419 L 3412 419 mt 3417 419 L 3417 419 mt 3423 420 L 3423 420 mt 3428 421 L 3428 421 mt 3434 421 L 3434 421 mt 3437 422 L 3437 422 mt 3439 422 L 3439 422 mt 3444 423 L 3444 423 mt 3450 424 L 3450 424 mt 3455 424 L 3455 424 mt 3460 425 L 3460 425 mt 3466 426 L 3466 426 mt 3469 426 L 3469 426 mt 3471 426 L 3471 426 mt 3476 427 L 34427 mt 3482 428 L 3482 428 mt 3487 429 L 3487 429 mt 3493 430 L 3493 430 mt 3498 430 L 3498 430 mt 3498 430 L 3498 430 mt 3503 431 L 3503 431 mt 3509 432 L 3509 432 mt 3514 433 L 3514 433 mt 3519 434 L 3519 434 mt 3524 435 L 3524 435 mt 3525 435 L 3525 435 mt 3530 436 L 3530 436 mt 3536 437 L 3536 437 mt 3541 438 L 3541 438 mt 3546 439 L 3546 439 mt 3548 439 L 3548 439 mt 3552 440 L 3552 440 mt 3557 441 L 3557 441 mt 3562 442 L 3562 442 mt 3568 443 L 3568 443 mt 3kIvI  *45 L 3579 445 mt 3584 446 L 3584 446 mt 3589 447 L 3589 447 mt 3591 447 L 3591 447 mt 3595 448 L 3595 448 mt 3600 449 L 3600 449 mt 3605 450 L 3605 450 mt 3611 451 L 3611 451 mt 3611 452 L 3611 452 mt 3616 453 L 3616 453 mt 3621 454 L 3621 454 mt 3627 455 L 3627 455 mt 3629 456 L 3629 456 mt 3632 457 L 3632 457 mt 3638 458 L 3638 458 mt 3643 459 L 3643 459 mt 3646 460 L 3646 460 mt 3648 460 L 3648 460 mt 3654 462 L 3B P S AH? E2I 9k[]a̢% v`! 9k[]a̢Z x=Pj@}3jmP -z~@,FH@r+~B~zLb}7K:RkVb!C2fj_t=Wt+ UXEtNb9+Y'ނ`G>R'I޲t l5ɓ_Hɏ։.ôL \&Wկ6Bn];mZbL˼piO$w"2Dd,@B Q5 mt 3740 487 L 3740 487 mt 3745 489 L 3745 489 mt 3747 490 L 3747 490 mt 3750 491 L 3750 491 mt 3756 492 L 3756 492 mt 3760 494 L 3760 494 mt 3761 494 L 3761 494 mt 3766 496 L 3766 496 mt 3772 498 L 3772 498 mt 3772 498 L 3772 498 mt 3777 500 L 3777 500 mt 3783 502 L 3783 502 mt 3783 502 L 3783 502 mt 3788 504 L 3788 504 mt 3793 506 L 3793 506 mt 3794 507 L 3794 507 mt 3799 508 L 3799 508 mt 3804 510 L 3804 510 mt 3805 511 L 3805 511 mt 3809 513 L  FMicrosoft Equation 3.0 DS Equation Equation.39q25 519 mt 3826 519 L 3826 519 mt 3831 522 L 3831 522 mt 3835 523 L 3835 523 mt 3836 524 L 3836 524 mt 3842 526 L 3842 526 mt 3845 528 L 3845 528 mt 3847 529 L 3847 529 mt 3852 531 L 3852 531 mt 3854 532 L 3854 532 mt 3858 534 L 3858 534 mt 3863 536 L 3863 536 mt 3863 536 L 3863 536 mt 3868 539 L 3868 539 mt 3872 540 L 3872 540 mt 3874 541 L 3874 541 mt 3879 544 L 3879 544 mt1 545 L 3881 545 mt 3885 546 L 3885 546 mt 3889 549 L 3889 549 mt 3890 549 L 3890 549 mt 3895 552 L 3895 552 mt 3898 553 L 3898 553 mt 3901 555 L 3901 555 mt 3906 557 L 3906 557 mt 3906 557 L 3906 557 mt 3911 560 L 3911 560 mt 3914 562 L 3914 562 mt 3917 563 L 3917 563 mt 3921 566 L 3921 566 mt 3922 566 L 3922 566 mt 3928 569 L 3928 569 mt 3929 570 L 3929 570 mt 3933 572 L 3933 572 mt 3937 574 L 3937 574 mt 3938 575 L 3938 575 mt 3944 578 L 3944 578 mt 3944 578 LIԊI  * 583 mt 3954 585 L 3954 585 mt 3958 587 L 3958 587 mt 3960 588 L 3960 588 mt 3965 591 L 3965 591 mt 3965 591 L 3965 591 mt 3970 594 L 3970 594 mt 3972 595 L 3972 595 mt 3976 598 L 3976 598 mt 3979 600 L 3979 600 mt 3981 601 L 3981 601 mt 3986 604 L 3986 604 mt 3987 605 L 3987 605 mt 3992 608 L 3992 608 mt 3992 608 L 3992 608 mt 3997 612 L 3997 612 mt 3998 612 L 3998 612 mt 4003 615 L 4003 615 mt 4005 617 L 4005 617 m S AI? F2LUֵ7 5{(v`! Uֵ7 5{Z xcdd``^ @bD"L1JE `x0 Yjl R A@v ^Դjx|K2B* R .ͤ `W01d++&10,D(03kf#c@+ss >׿Q ..)I)$5N @dd`3]YDd |B R S AJ? 662 L 4067 662 mt 4069 663 L 4069 663 mt 4073 666 L 4073 666 mt 4074 667 L 4074 667 mt 4078 670 L 4078 670 mt 4080 672 L 4080 672 mt 4083 675 L 4083 675 mt 4085 676 L 4085 676 mt 4089 679 L 4089 679 mt 4090 680 L 4090 680 mt 4094 683 L 4094 683 mt 4095 684 L 4095 684 mt 4099 688 L 4099 688 mt 4100 689 L 4100 689 mt 4105 692 L 4105 692 mt 4105 693 L 4105 693 mt 4110 697 L 4110 697 mt 4110 697 L 4110 697 mt 4115 701 L 4115 701 mt 4115 702 L 4115 702  FMicrosoft Equation 3.0 DS Equation Equation.39q 4130 714 L 4130 714 mt 4132 716 L 4132 716 mt 4134 718 L 4134 718 mt 4137 721 L 4137 721 mt 4139 722 L 4139 722 mt 4142 726 L 4142 726 mt 4143 727 L 4143 727 mt 4148 730 L 4148 730 mt 4148 731 L 4148 731 mt 4153 735 L 4153 735 mt 4153 736 L 4153 736 mt 4157 739 L 4157 739 mt 4158 741 L 4158 741 mt 4161 744 L 4161 744 mt 4164 746 L 4164 746 mt 4166 748 L 4166 748 mt 4169 751169 751 mt 4170 752 L 4170 752 mt 4175 756 L 4175 756 mt 4175 756 L 4175 756 mt 4179 760 L 4179 760 mt 4180 762 L 4180 762 mt 4183 765 L 4183 765 mt 4185 767 L 4185 767 mt 4187 769 L 4187 769 mt 4191 772 L 4191 772 mt 4191 773 L 4191 773 mt 4196 777 L 4196 777 mt 4196 778 L 4196 778 mt 4200 782 L 4200 782 mt 4201 783 L 4201 783 mt 4204 786 L 4204 786 mt 4207 789 L 4207 789 mt 4208 790 L 4208 790 mt 4212 794 L 4212 794 mt 4212 795 L 4212 795 mt 4216 799 L 4216 799II w T " xx L(x * , * , * )w>0,31 815 mt 4234 818 L 4234 818 mt 4235 820 L 4235 820 mt 4239 824 L 4239 824 mt 4239 824 L 4239 824 mt 4242 828 L 4242 828 mt 4244 830 L 4244 830 mt 4246 832 L 4246 832 mt 4250 837 L 4250 837 mt 4250 837 L 4250 837 mt 4253 841 L 4253 841 mt 4255 843 L 4255 843 mt 4257 845 L 4257 845 mt 4260 849 L 4260 849 mt 4261 84 G2%dS's6v`!%dS's6H`0exJPgפ( J Qhl&]3q+"7$9FP9?'3@WN xUe)YYY %ԤaeJR|fj+J_4eoOm4 ?\ 3A.ɪz_TMsO,^Ѽ%Zb^XLI6r ZWG(=kg/\.&$>;jq 8S`Wd~Het>lŒy9˼~<߾<~& v*#1I j,0ДN}ÂН65^qC=gD PuDd5 904 mt 4308 909 L 4308 909 mt 4309 909 L 4309 909 mt 4312 913 L 4312 913 mt 4314 916 L 4314 916 mt 4315 917 L 4315 917 mt 4318 921 L 4318 921 mt 4320 923 L 4320 923 mt 4321 926 L 4321 926 mt 4324 930 L 4324 930 mt 4325 930 L 4325 930 mt 4327 934 L 4327 934 mt 4330 938 L 4330 938 mt 4331 938 L 4331 938 mt 4334 942 L 4334 942 mt 4336 945 L 4336 945 mt 4337 947 L 4337 947 mt 4340 951 L 4340 951 mt 4341 953 L 4341 953 mt 4343 955 L 4343 955 mt 4346 9 FMicrosoft Equation 3.0 DS Equation Equation.39q L 4352 968 mt 4355 972 L 4355 972 mt 4357 975 L 4357 975 mt 4358 976 L 4358 976 mt 4361 981 L 4361 981 mt 4362 983 L 4362 983 mt 4364 985 L 4364 985 mt 4366 989 L 4366 989 mt 4368 991 L 4368 991 mt 4369 993 L 4369 993 mt 4372 997 L 4372 997 mt 4373 999 L 4373 999 mt 4375 1002 L 4375 1002 mt 4378 1006 L 4378 1006 mt 4379 1007 L 4379 1007 mt 4381 1010 L 4381 1010 mt 4383 1014 L 4383 10t 4384 1015 L 4384 1015 mt 4386 1019 L 4386 1019 mt 4389 1023 L 4389 1023 mt 4389 1024 L 4389 1024 mt 4392 1027 L 4392 1027 mt 4394 1031 L 4394 1031 mt 4395 1032 L 4395 1032 mt 4397 1036 L 4397 1036 mt 4400 1040 L 4400 1040 mt 4400 1040 L 4400 1040 mt 4402 1044 L 4402 1044 mt 4405 1048 L 4405 1048 mt 4405 1049 L 4405 1049 mt 4408 1052 L 4408 1052 mt 4410 1057 L 4410 1057 mt 4411 1058 L 4411 1058 mt 4413 1061 L 4413 1061 mt 4415 1065 L 4415 1065 mt 4416 1066 L 4416 1066 mt 4418 1069 L 4418 1069 mt 4421 14IԊI w4423 1078 L 4423 1078 mt 4426 1082 L 4426 1082 mt 4427 1084 L 4427 1084 mt 4428 1086 L 4428 1086 mt 4431 1091 L 4431 1091 mt 4432 1093 L 4432 1093 mt 4433 1095 L 4433 1095 mt 4436 1099 L 4436 1099 mt 4438 1102 L 4438 1102 mt 4438 1103 L 4438 1103 mt 4441 1108 L 4441 1108 mt 4443 1112 L 4443 1112 mt 4443 1112 L 4443 1112 mt 4445 1116 L 4445 1116 mt 4448 1120 L 4448 1120 mt 4448 1121 L 4448 1121 mt 4450 1124 L 4450 1124 mt 4453 1129 L 4453 1129 mt 4454 1131 L 4454 1B S S AK? H2%4!Y FOD&v`!4!Y FOD`R!x5O ` , NЭ>A_@ءCUPur| }#w/B 4ה販,)B\U"S*2gu.oT~*#'C*_r(Ss\,l75 Ëir|>x'|w$/lPasmM:ө-1O69Kߝ2%+&&DdhB T S פI nI maxb T subjecttoA T +s=c,se"0, FMicrosoft Equation 3.0 DS Equation Equation.39q FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native CompObjfObjInfo89517213F`A AOle Equation Native CompObjfH|IlI "h(x * ) T w=0maxb T subjecttoA T +s=c,se"0,  FMicrosoft Equation 3.0 DS Equation Equation.39qAL? I2iQg#vA=Hlv`!diQg#vA=H @ |2xRJAf.w xXA[E̽@ $_ `c9,B֝fPovof)5)[ Y5&1F#MZˮcFj 6`k%]NgA'-zp ~B@yOγar۷(ٳM1x\SۮCbځti,ҮN{++yHM~[lDp?wh~8{H_ZE blAjTh7Va,Dd| nI maxb T subjecttoA T +s=c,se"0,  FMicrosoft Equation 3.0 DS Equation Equation.39qפI nI maxb T subjecttoA FMicrosoft Equation 3.0 DS Equation Equation.39qfObjInfoEquation Native CompObj|f T +s=c,se"0, FMicrosoft Equation 3.0 DS Equation Equation.39q FMicrosoft Equation 3.0 DS Equation Equation.39qפI nI maxb TII "c i (x * ) T w=0AOle CompObjfObjInfoB U S AM? J2 x ۀ^rv`!j x ۀ^` `08xRAJP}3imV ( QK7ՅR%hXڝG@p)x)ߏ 5aȼof 0OYgĘH#EQhAKfpϋ:3e& '"i(qǃHT}ܓ`q;KQڏ;Y x3J6*ߝ`HgUCrd7vvv u왤 jB+6I?fS}< 774DdxB 6 S A3? 0 2690 mt 4780 2695 L 4780 2695 mt 4779 2699 L 4779 2699 mt 4779 2703 L 4779 2703 mt 4779 2707 L 4779 2707 mt 4779 2712 L 4779 2712 mt 4778 2716 L 4778 2716 mt 4778 2720 L 4778 2720 mt 4778 2724 L 4778 2724 mt 4777 2729 L 4777 2729 mt 4777 2733 L 4777 2733 mt 4777 2737 L 4777 2737 mt 4777 2741 L 4777 2741 mt 4776 2745 L 4776 2745 mt 4776 2748 L 4776 2748 mt 4776 2750 L 4776 2750 mt 4776 2754 L 4776 2754 mt 4775 2758 L 4775 2758 mt 4775 2762 L 4775 2762 mt 4774 2767 L 4774 2767 mt 4774 27 FMicrosoft Equation 3.0 DS Equation Equation.39q L 4773 2788 mt 4772 2792 L 4772 2792 mt 4772 2796 L 4772 2796 mt 4771 2801 L 4771 2801 mt 4771 2805 L 4771 2805 mt 4771 2809 L 4771 2809 mt 4771 2809 L 4771 2809 mt 4770 2813 L 4770 2813 mt 4770 2817 L 4770 2817 mt 4769 2822 L 4769 2822 mt 4769 2826 L 4769 2826 mt 4768 2830 L 4768 2830 mt 4768 2834 L 4768 2834 mt 4768 2839 L 4768 2839 mt 4767 2843 L 4767 2843 mt 4767 2847 L 4767 2847 mt 4766 2851 L 4766 28t 4766 2856 L 4766 2856 mt 4765 2860 L 4765 2860 mt 4765 2860 L 4765 2860 mt 4765 2864 L 4765 2864 mt 4764 2868 L 4764 2868 mt 4764 2872 L 4764 2872 mt 4763 2877 L 4763 2877 mt 4763 2881 L 4763 2881 mt 4762 2885 L 4762 2885 mt 4762 2889 L 4762 2889 mt 4761 2894 L 4761 2894 mt 4761 2898 L 4761 2898 mt 4760 2902 L 4760 2902 mt 4760 2904 L 4760 2904 mt 4760 2906 L 4760 2906 mt 4759 2911 L 4759 2911 mt 4758 2915 L 4758 2915 mt 4758 2919 L 4758 2919 mt 4757 2923 L 4757 2923 mt 4757 2927 L 4757 2927 mt 4756 2 I@I 'min d "f(x k ) T d+12d T H k d,subjecttoc(x k )+"c(x k ) T dd"0,h(x k )+"h(x k ) T d=0,I2995 mt 4746 2999 L 4746 2999 mt 4745 3004 L 4745 3004 mt 4745 3008 L 4745 3M2g09 JߓZzv`!rg09 JߓZT@$' @xAhPg&Mv7mSWi)XS zpWD!`[#t x۫!x"ţ 饈P=^Z47 gdC(X/H`OU!R@̝)otPIuu88-H E1*j{tlsd2gx{uSqfAc4Ӡ{ȗg|˽W|Mq}/]`Vguz8+Ug ӊ<dJQ"AZzsgj*>{E:o.ˆk-m 鼆zvM>k N!;|ja >8LoCF@XΥDdhhB B S A>? N2:} N)&v`!} N)&d@@||xcdd`` @bD"L1JE `xPRcgb x@πjx|4716 3160 mt 4715 3164 L 4715 3164 mt 4714 3169 L 4714 3169 mt 4713 3173 L 4713 3173 mt 4712 3177 L 4712 3177 mt 4712 3179 L 4712 3179 mt 4711 3181 L 4711 3181 mt 4710 3186 L 4710 3186 mt 4709 3190 L 4709 3190 mt 4708 3194 L 4708 3194 mt 4707 3198 L 4707 3198 mt 4706 3203 L 4706 3203 mt 4706 3203 L 4706 3203 mt 4705 3207 L 4705 3207 mt 4704 3211 L 4704 3211 mt 4703 3215 L 4703 3215 mt 4702 3220 L 4702 3220 mt 4701 3224 L 4701 3224 mt 4701 3225 L 4701 3225 mt 4700 3228 L 4700 3228 mt 4699 FMicrosoft Equation 3.0 DS Equation Equation.39q247 L 4695 3247 mt 4695 3249 L 4695 3249 mt 4694 3253 L 4694 3253 mt 4693 3258 L 4693 3258 mt 4692 3262 L 4692 3262 mt 4691 3266 L 4691 3266 mt 4690 3268 L 4690 3268 mt 4690 3270 L 4690 3270 mt 4688 3275 L 4688 3275 mt 4687 3279 L 4687 3279 mt 4686 3283 L 4686 3283 mt 4685 3287 L 4685 3287 mt 4685 3289 L 4685 3289 mt 4684 3291 L 4684 3291 mt 4683 3296 L 4683 3296 mt 4682 3300 L 4682 3300 mt 4681 3304 L 46814 mt 4679 3308 L 4679 3308 mt 4679 3309 L 4679 3309 mt 4678 3313 L 4678 3313 mt 4677 3317 L 4677 3317 mt 4676 3321 L 4676 3321 mt 4675 3325 L 4675 3325 mt 4674 3328 L 4674 3328 mt 4674 3330 L 4674 3330 mt 4672 3334 L 4672 3334 mt 4671 3338 L 4671 3338 mt 4670 3342 L 4670 3342 mt 4669 3346 L 4669 3346 mt 4669 3347 L 4669 3347 mt 4668 3351 L 4668 3351 mt 4666 3355 L 4666 3355 mt 4665 3359 L 4665 3359 mt 4664 3363 L 4664 3363 mt 4663 3365 L 4663 3365 mt 4663 3368 L 4663 3368 mt 4661 3372 L 4661 3372 mt 466\II H k L 4658 3383 mt 4657 3385 L 4657 3385 mt 4656 3389 L 4656 3389 mt 4655 3393 L 4655 3393 mt 4654 3397 L 4654 3397 mt 4652 3401 L 4652 3401 mt 4652 3402 L 4652 3402 mt 4651 3406 L 4651 3406 mt 4650 3410 L 4650 3410 mt 4648 3414 L 4648 3414 mt 4647 3418 L 4647 3418 mt 4647 3418 L 4647 3418 mt 4646 3423 L 4646 3423 mt 4644 3427 L 4644 3427 mt 4643 3431 L 4643 3431 mt 4642 3435 L 4642 3435 mt 4642 3435 L 4642 3435 mt 4640 3440 L 4640 3440 mt 4639 3444 L 463K2B* R.ͤ XB2sSRsN\ F\j&4#F>Mj`tc` Fy`=`321)WBB * `ux`6DdB C S A?? O2'/6QSn%-~|v`!/6QSn%-~|`!Hx5O A q!buX)j?h~ .hq*܁ YX=k6 !L2!@ ф{E ]9Hbk"}j)2dUjRuKua.՜Sdǽͧ.mt 4614 3516 L 4614 3516 mt 4612 3520 L 4612 3520 mt 4611 3524 L 4611 3524 mt 4609 3528 L 4609 3528 mt 4609 3528 L 4609 3528 mt 4608 3533 L 4608 3533 mt 4606 3537 L 4606 3537 mt 4605 3541 L 4605 3541 mt 4604 3543 L 4604 3543 mt 4603 3545 L 4603 3545 mt 4601 3550 L 4601 3550 mt 4600 3554 L 4600 3554 mt 4599 3557 L 4599 3557 mt 4598 3558 L 4598 3558 mt 4597 3562 L 4597 3562 mt 4595 3567 L 4595 3567 mt 4593 3571 L 4593 3571 mt 4593 3571 L 4593 3571 mt 4592 3575 L 4592 3575 mt 4590 3579 L 45 FMicrosoft Equation 3.0 DS Equation Equation.39q 3592 mt 4583 3596 L 4583 3596 mt 4583 3598 L 4583 3598 mt 4582 3600 L 4582 3600 mt 4580 3605 L 4580 3605 mt 4578 3609 L 4578 3609 mt 4577 3612 L 4577 3612 mt 4577 3613 L 4577 3613 mt 4575 3617 L 4575 3617 mt 4573 3622 L 4573 3622 mt 4572 3625 L 4572 3625 mt 4571 3626 L 4571 3626 mt 4570 3630 L 4570 3630 mt 4568 3634 L 4568 3634 mt 4567 3638 L 4567 3638 mt 4566 3639 L 4566 3639 mt 4564 3643 L 4564 3643 mt 43647 L 4563 3647 mt 4561 3650 L 4561 3650 mt 4561 3651 L 4561 3651 mt 4559 3655 L 4559 3655 mt 4557 3660 L 4557 3660 mt 4556 3663 L 4556 3663 mt 4555 3664 L 4555 3664 mt 4553 3668 L 4553 3668 mt 4552 3672 L 4552 3672 mt 4550 3675 L 4550 3675 mt 4550 3677 L 4550 3677 mt 4548 3681 L 4548 3681 mt 4546 3685 L 4546 3685 mt 4545 3687 L 4545 3687 mt 4544 3689 L 4544 3689 mt 4542 3694 L 4542 3694 mt 4540 3698 L 4540 3698 mt 4540 3699 L 4540 3699 mt 4538 3702 L 4538 3702 mt 4536 3706 L 4536 3706 mt 4534 3710 L 4kIvI d15 L 4533 3715 mt 4531 3719 L 4531 3719 mt 4529 3722 L 4529 3722 mt 4529 3723 L 4529 3723 mt 4527 3727 L 4527 3727 mt 4525 3732 L 4525 3732 mt 4524 3734 L 4524 3734 mt 4523 3736 L 4523 3736 mt 4521 3740 L 4521 3740 mt 4519 3744 L 4519 3744 mt 4518 3745 L 4518 3745 mt 4517 3749 L 4517 3749 mt 4515 3753 L 4515 3753 mt 4513 3756 L 4513 3756 mt 4512 3757 L 4512 3757 mt 4510 3761 L 4510 3761 mt 4508 3766 L 4508 3766 mt 4507 3767 L 4507 3767 mt 4506 3770 L 4506 3770 mt 6I9D]g'ɣ 1w n9N Qn3?I~wd/q*Dd,@B D S A@? P2:|v1?b0=v`!|v1?b0=d xPKA}UCb"9,&!2Hf$.`cc(`a'Ѫ˰I_Ur@Q]).iQ"E1H)f+0F%"0VWWRG1$+hk'{F@$?.$Mu)w;^:t9::.2z\m37 L 4471 3837 mt 4470 3840 L 4470 3840 mt 4469 3842 L 4469 3842 mt 4467 3846 L 4467 3846 mt 4465 3850 L 4465 3850 mt 4464 3850 L 4464 3850 mt 4462 3854 L 4462 3854 mt 4460 3859 L 4460 3859 mt 4459 3860 L 4459 3860 mt 4457 3863 L 4457 3863 mt 4455 3867 L 4455 3867 mt 4454 3869 L 4454 3869 mt 4453 3871 L 4453 3871 mt 4450 3876 L 4450 3876 mt 4448 3879 L 4448 3879 mt 4448 3880 L 4448 3880 mt 4445 3884 L 4445 3884 mt 4443 3888 L 4443 3888 mt 4443 3888 L 4443 3888 mt 4441 3892 L 4441 3892 mt FMicrosoft Equation 3.0 DS Equation Equation.39q432 3907 L 4432 3907 mt 4431 3909 L 4431 3909 mt 4428 3914 L 4428 3914 mt 4427 3916 L 4427 3916 mt 4426 3918 L 4426 3918 mt 4423 3922 L 4423 3922 mt 4422 3925 L 4422 3925 mt 4421 3926 L 4421 3926 mt 4418 3931 L 4418 3931 mt 4416 3934 L 4416 3934 mt 4415 3935 L 4415 3935 mt 4413 3939 L 4413 3939 mt 4411 3942 L 4411 3942 mt 4410 3943 L 4410 3943 mt 4408 3948 L 4408 3948 mt 4405 3951 L 4405 3951 mt 4405 3952 L5 3952 mt 4402 3956 L 4402 3956 mt 4400 3960 L 4400 3960 mt 4400 3960 L 4400 3960 mt 4397 3964 L 4397 3964 mt 4395 3968 L 4395 3968 mt 4394 3969 L 4394 3969 mt 4392 3973 L 4392 3973 mt 4389 3976 L 4389 3976 mt 4389 3977 L 4389 3977 mt 4386 3981 L 4386 3981 mt 4384 3985 L 4384 3985 mt 4383 3986 L 4383 3986 mt 4381 3990 L 4381 3990 mt 4379 3993 L 4379 3993 mt 4378 3994 L 4378 3994 mt 4375 3998 L 4375 3998 mt 4373 4001 L 4373 4001 mt 4372 4003 L 4372 4003 mt 4369 4007 L 4369 4007 mt 4368 4009 L 4368 4009 mkIvI x k 4017 L 4362 4017 mt 4361 4019 L 4361 4019 mt 4358 4024 L 4358 4024 mt 4357 4025 L 4357 4025 mt 4355 4028 L 4355 4028 mt 4352 4032 L 4352 4032 mt 4352 4032 L 4352 4032 mt 4349 4036 L 4349 4036 mt 4346 4040 L 4346 4040 mt 4346 4041 L 4346 4041 mt 4343 4045 L 4343 4045 mt 4341 4047 L 4341 4047 mt 4340 4049 L 4340 4049 mt 4337 4053 L 4337 4053 mt 4336 4055 L 4336 4055 mt 4334 4058 L 4334 4058 mt 4331 4062 L 4331 4062 mt 4330 4062 L 4330 4062 mt 4327 4066 >?fS}< 77DdhhB E S AA? Q2:6rD[mK#Tv`!6rD[mK#d@@||xcdd`` @bD"L1JE `xPRcgb x@πjx|K2B* R.ͤ XB2sSRsA.Jsi#]`Pk#O&501x0pA]#< `+KRs!!rLFf:+6s4Dd|4121 mt 4282 4125 L 4282 4125 mt 4282 4125 L 4282 4125 mt 4278 4129 L 4278 4129 mt 4277 4132 L 4277 4132 mt 4275 4134 L 4275 4134 mt 4271 4138 L 4271 4138 mt 4271 4138 L 4271 4138 mt 4268 4142 L 4268 4142 mt 4266 4145 L 4266 4145 mt 4264 4146 L 4264 4146 mt 4261 4151 L 4261 4151 mt 4260 4151 L 4260 4151 mt 4257 4155 L 4257 4155 mt 4255 4157 L 4255 4157 mt 4253 4159 L 4253 4159 mt 4250 4163 L 4250 4163 mt 4250 4163 L 4250 4163 mt 4246 4168 L 4246 4168 mt 4244 4170 L 4244 4170 mt 4242 4172 FMicrosoft Equation 3.0 DS Equation Equation.39q 4234 4182 mt 4231 4185 L 4231 4185 mt 4228 4188 L 4228 4188 mt 4227 4189 L 4227 4189 mt 4223 4193 L 4223 4193 mt 4223 4194 L 4223 4194 mt 4220 4197 L 4220 4197 mt 4217 4199 L 4217 4199 mt 4216 4201 L 4216 4201 mt 4212 4205 L 4212 4205 mt 4212 4206 L 4212 4206 mt 4208 4210 L 4208 4210 mt 4207 4211 L 4207 4211 mt 4204 4214 L 4204 4214 mt 4201 4217 L 4201 4217 mt 4200 4218 L 4200 4218 mt 4196 4222 L 4196 42224196 4223 L 4196 4223 mt 4191 4227 L 4191 4227 mt 4191 4228 L 4191 4228 mt 4187 4231 L 4187 4231 mt 4185 4233 L 4185 4233 mt 4183 4235 L 4183 4235 mt 4180 4238 L 4180 4238 mt 4179 4240 L 4179 4240 mt 4175 4244 L 4175 4244 mt 4175 4244 L 4175 4244 mt 4170 4248 L 4170 4248 mt 4169 4249 L 4169 4249 mt 4166 4252 L 4166 4252 mt 4164 4254 L 4164 4254 mt 4161 4256 L 4161 4256 mt 4158 4259 L 4158 4259 mt 4157 4261 L 4157 4261 mt 4153 4264 L 4153 4264 mt 4153 4265 L 4153 4265 mt 4148 4269 L 4148 4269 mt 4148 427kIvI H k42 4274 mt 4139 4278 L 4139 4278 mt 4137 4279 L 4137 4279 mt 4134 4282 L 4134 4282 mt 4132 4284 L 4132 4284 mt 4130 4286 L 4130 4286 mt 4126 4289 L 4126 4289 mt 4125 4290 L 4125 4290 mt 4121 4294 L 4121 4294 mt 4120 4295 L 4120 4295 mt 4115 4298 L 4115 4298 mt 4115 4299 L 4115 4299 mt 4110 4303 L 4110 4303 mt 4110 4303 L 4110 4303 mt 4105 4307 L 4105 4307 mt 4105 4308 L 4105 4308 mt 4100 4311 L 4100 4311 mt 4099 4312 L 4099 4312 mt 4095 4316 L 4095 4314 4317 mt 4090 4320 L 4090 4320 mt 4089 4321 L 4089 4321 mt 4085 4324 L 4085 4324 mt 4083 4325 L 4083 4325 mt 4080 4328 L 4080 4328 mt 4078 4330 L 4078 4330 mt 4074 4333 L 4074 4333 mt 4073 4334 L 4073 4334 mt 4069 4337 L 4069 4337 mt 4067 4338 L 4067 4338 mt 4063 4341 L 4063 4341 mt 4062 4342 L 4062 4342 mt 4058 4345 L 4058 4345 mt 4056 4346 L 4056 4346 mt 4052 4350 L 4052 4350 mt 4051 4351 L 4051 4351 mt 4047 4354 L 4047 4354 mt 4046 4355 L 4046 4355 mt 4041 4358 L 4041 4358 mt 4040 43 FMicrosoft Equation 3.0 DS Equation Equation.39q L 4029 4367 mt 4024 4370 L 4024 4370 mt 4023 4371 L 4023 4371 mt 4019 4374 L 4019 4374 mt 4017 4375 L 4017 4375 mt 4013 4378 L 4013 4378 mt 4011 4379 L 4011 4379 mt 4008 4381 L 4008 4381 mt 4005 4383 L 4005 4383 mt 4003 4385 L 4003 4385 mt 3998 4388 L 3998 4388 mt 3997 4388 L 3997 4388 mt 3992 4392 L 3992 4392 mt 3992 4392 L 3992 4392 mt 3987 4395 L 3987 4395 mt 3986 4396 L 3986 4396 mt 3981 4399 L 3981 43t 3979 4400 L 3979 4400 mt 3976 4402 L 3976 4402 mt 3972 4405 L 3972 4405 mt 3970 4406 L 3970 4406 mt 3965 4409 L 3965 4409 mt 3965 4409 L 3965 4409 mt 3960 4412 L 3960 4412 mt 3958 4413 L 3958 4413 mt 3954 4415 L 3954 4415 mt 3951 4417 L 3951 4417 mt 3949 4419 L 3949 4419 mt 3944 4422 L 3944 4422 mt 3944 4422 L 3944 4422 mt 3938 4425 L 3938 4425 mt 3937 4426 L 3937 4426 mt 3933 4428 L 3933 4428 mt 3929 4430 L 3929 4430 mt 3928 4431 L 3928 4431 mt 3922 4434 L 3922 4434 mt 3921 4434 L 3921 4434 mt 3917 4pkIvI " xx L(x k , k , k ), mt 3898 4447 L 3898 4447 mt 3895 4448 L 3895 4448 mt 3890 4451 L 3890 4451 mt 3889 4451 L 3889 4451 mt 3885 4454 L 3885 4454 mt 3881 4455 L 3881 4455 mt 3879 4456 L 3879 4456 mt 3874 4459 L 3874 4459 mt 3872 4460 L 3872 4460 mt 3868 4461 L 3868 4461 mt 3863 4464 L 3863 4464 mt 3863 4464 L 3863 4464 mt 3858 4466 L 3858 4466 mt 3854 4468 L 3854 4468 mt 3852 4469 L 3852 4B F S AB? R26h!E#ߟdSz$v`!r6h!E#ߟdS `(0@xRJ@fߴ`DB]}b7K 7;?, 8g0֠*kBeX!}&]?fS}< 77Dd3697 4526 mt 3694 4527 L 3694 4527 mt 3691 4528 L 3691 4528 mt 3686 4530 L 3686 4530 mt 3681 4531 L 3681 4531 mt 3679 4532 L 3679 4532 mt 3675 4532 L 3675 4532 mt 3670 4534 L 3670 4534 mt 3664 4535 L 3664 4535 mt 3663 4536 L 3663 4536 mt 3659 4537 L 3659 4537 mt 3654 4538 L 3654 4538 mt 3648 4540 L 3648 4540 mt 3646 4540 L 3646 4540 mt 3643 4541 L 3643 4541 mt 3638 4542 L 3638 4542 mt 3632 4543 L 3632 4543 mt 3629 4544 L 3629 4544 mt 3627 4545 L 3627 4545 mt 3621 4546 L 3621 4546 mt 3616 FMicrosoft Equation 3.0 DS Equation Equation.39q551 L 3600 4551 mt 3595 4552 L 3595 4552 mt 3591 4553 L 3591 4553 mt 3589 4553 L 3589 4553 mt 3584 4554 L 3584 4554 mt 3579 4555 L 3579 4555 mt 3573 4556 L 3573 4556 mt 3570 4557 L 3570 4557 mt 3568 4557 L 3568 4557 mt 3562 4558 L 3562 4558 mt 3557 4559 L 3557 4559 mt 3552 4560 L 3552 4560 mt 3548 4561 L 3548 4561 mt 3546 4561 L 3546 4561 mt 3541 4562 L 3541 4562 mt 3536 4563 L 3536 4563 mt 3530 4564 L 35304 mt 3525 4565 L 3525 4565 mt 3524 4565 L 3524 4565 mt 3519 4566 L 3519 4566 mt 3514 4567 L 3514 4567 mt 3509 4568 L 3509 4568 mt 3503 4569 L 3503 4569 mt 3498 4570 L 3498 4570 mt 3498 4570 L 3498 4570 mt 3493 4570 L 3493 4570 mt 3487 4571 L 3487 4571 mt 3482 4572 L 3482 4572 mt 3476 4573 L 3476 4573 mt 3471 4574 L 3471 4574 mt 3469 4574 L 3469 4574 mt 3466 4574 L 3466 4574 mt 3460 4575 L 3460 4575 mt 3455 4576 L 3455 4576 mt 3450 4576 L 3450 4576 mt 3444 4577 L 3444 4577 mt 3439 4578 L 3439 4578 mt 343kIvI x k L 3428 4579 mt 3423 4580 L 3423 4580 mt 3417 4581 L 3417 4581 mt 3412 4581 L 3412 4581 mt 3407 4582 L 3407 4582 mt 3402 4582 L 3402 4582 mt 3401 4582 L 3401 4582 mt 3396 4583 L 3396 4583 mt 3391 4584 L 3391 4584 mt 3385 4584 L 3385 4584 mt 3380 4585 L 3380 4585 mt 3374 4585 L 3374 4585 mt 3369 4586 L 3369 4586 mt 3364 4586 L 3364 4586 mt 3361 4587 L 3361 4587 mt 3358 4587 L 3358 4587 mt 3353 4587 L 3353 4587 mt 3348 4588 L 3348 4588 mt 3342 4588 L 334L 3337 4589 mt 3331 4589 L 3331 4589 mt 3326 4590 L 3326 4590 mt 3321 4590 L 3321 4590 mt 3315 4591 L 3315 4591 mt 3315 4591 L 3315 4591 mt 3310 4591 L 3310 4591 mt 3305 4592 L 3305 4592 mt 3299 4592 L 3299 4592 mt 3294 4592 L 3294 4592 mt 3289 4593 L 3289 4593 mt 3283 4593 L 3283 4593 mt 3278 4594 L 3278 4594 mt 3272 4594 L 3272 4594 mt 3267 4594 L 3267 4594 mt 3262 4595 L 3262 4595 mt 3259 4595 L 3259 4595 mt 3256 4595 L 3256 4595 mt 3251 4596 L 3251 4596 mt 3246 4596 L 3246 4596 mt 32 FMicrosoft Equation 3.0 DS Equation Equation.39q 4598 L 3219 4598 mt 3213 4598 L 3213 4598 mt 3208 4598 L 3208 4598 mt 3203 4599 L 3203 4599 mt 3197 4599 L 3197 4599 mt 3192 4599 L 3192 4599 mt 3190 4599 L 3190 4599 mt 3187 4599 L 3187 4599 mt 3181 4600 L 3181 4600 mt 3176 4600 L 3176 4600 mt 3170 4600 L 3170 4600 mt 3165 4601 L 3165 4601 mt 3160 4601 L 3160 4601 mt 3154 4601 L 3154 4601 mt 3149 4601 L 3149 4601 mt 3144 4602 L 3144 4602 mt 3138 4602 L 31602 mt 3133 4602 L 3133 4602 mt 3127 4602 L 3127 4602 mt 3122 4603 L 3122 4603 mt 3117 4603 L 3117 4603 mt 3111 4603 L 3111 4603 mt 3106 4603 L 3106 4603 mt 3101 4604 L 3101 4604 mt 3101 4604 L 3101 4604 mt 3095 4604 L 3095 4604 mt 3090 4604 L 3090 4604 mt 3084 4604 L 3084 4604 mt 3079 4604 L 3079 4604 mt 3074 4604 L 3074 4604 mt 3068 4605 L 3068 4605 mt 3063 4605 L 3063 4605 mt 3058 4605 L 3058 4605 mt 3052 4605 L 3052 4605 mt 3047 4605 L 3047 4605 mt 3042 4605 L 3042 4605 mt 3036 4606 L 3036 4606 mt 30I@I  k , k15 4606 L 3015 4606 mt 3009 4607 L 3009 4607 mt 3004 4607 L 3004 4607 mt 2999 4607 L 2999 4607 mt 2993 4607 L 2993 4607 mt 2988 4607 L 2988 4607 mt 2982 4607 L 2982 4607 mt 2977 4608 L 2977 4608 mt 2972 4608 L 2972 4608 mt 2971 4608 L 2971 4608 mt 2966 4608 L 2966 4608 mt 2961 4608 L 2961 4608 mt 2956 4608 L 2956 4608 mt 2950 4608 L 2950 4608 mt 2945 4608 L 2945 4608 mt 2940 4608 L 2940 4608 mt 2934 4608 L 2934 4608 mt 2929 4609 L 2hB H S AC? T2d$pA1<U@(v`!8$pA1<U@@2|xcdd``~ @bD"L1JE `xX,56~) M @ k;`f` UXRYvo&`0L` ZZ]46 R 3B`!I9 Ls>7o&wp u!<.7i;e: \ k=pedbR ,.IͅZf:6C^Ddhh1 mt 2811 4611 L 2811 4611 mt 2805 4611 L 2805 4611 mt 2800 4611 L 2800 4611 mt 2795 4611 L 2795 4611 mt 2789 4611 L 2789 4611 mt 2784 4611 L 2784 4611 mt 2778 4611 L 2778 4611 mt 2773 4611 L 2773 4611 mt 2768 4611 L 2768 4611 mt 2762 4612 L 2762 4612 mt 2757 4612 L 2757 4612 mt 2752 4612 L 2752 4612 mt 2746 4612 L 2746 4612 mt 2741 4612 L 2741 4612 mt 2737 4612 L /c14 { 0.125000 1.000000 0.937500 sr} bdef c14 3362 388 mt 3364 388 L 3364 388 mt 3369 388 L 3369 388 mt 3374 388 L 337 FMicrosoft Equation 3.0 DS Equation Equation.39q 389 mt 3401 389 L 3401 389 mt 3407 389 L 3407 389 mt 3412 389 L 3412 389 mt 3417 389 L 3417 389 mt 3423 389 L 3423 389 mt 3428 389 L 3428 389 mt 3434 390 L 3434 390 mt 3439 390 L 3439 390 mt 3444 390 L 3444 390 mt 3450 390 L 3450 390 mt 3455 390 L 3455 390 mt 3460 390 L 3460 390 mt 3466 390 L 3466 390 mt 3471 390 L 3471 390 mt 3476 390 L 3476 390 mt 3482 391 L 3482 391 mt 34391 L 3487 391 mt 3493 391 L 3493 391 mt 3498 391 L 3498 391 mt 3503 391 L 3503 391 mt 3509 391 L 3509 391 mt 3514 391 L 3514 391 mt 3519 391 L 3519 391 mt 3525 392 L 3525 392 mt 3530 392 L 3530 392 mt 3536 392 L 3536 392 mt 3541 392 L 3541 392 mt 3546 392 L 3546 392 mt 3552 392 L 3552 392 mt 3556 392 L 3556 392 mt 3557 392 L 3557 392 mt 3562 392 L 3562 392 mt 3568 393 L 3568 393 mt 3573 393 L 3573 393 mt 3579 393 L 3579 393 mt 3584 393 L 3584 393 mt 3589 393 L 35kIvI H k4 mt 3605 394 L 3605 394 mt 3611 394 L 3611 394 mt 3616 395 L 3616 395 mt 3621 395 L 3621 395 mt 3627 395 L 3627 395 mt 3632 395 L 3632 395 mt 3638 395 L 3638 395 mt 3643 396 L 3643 396 mt 3648 396 L 3648 396 mt 3654 396 L 3654 396 mt 3659 396 L 3659 396 mt 3664 396 L 3664 396 mt 3666 396 L 3666 396 mt 3670 397 L 3670 397 mt 3675 397 L 3675 397 mt 3681 397 L 3681 397 mt 3686 398 L 3686 398 mt 3691 398 L 3691 398 mt 3B I S AA? U2:6rD[mK#"v`!6rD[mK#d@@||xcdd`` @bD"L1JE `xPRcgb x@πjx|K2B* R.ͤ XB2sSRsA.Jsi#]`Pk#O&501x0pA]#< `+KRs!!rLFf:+6sDdhhB  FMicrosoft Equation 3.0 DS Equation Equation.39qkIvI H kJ S AA? V2:6rD[mK#v`!6rD[mK#d@@||xcdd`` @bD"L1JE `xPRcgb x@πjx|K2B* R.ͤ XB2sSRsA.Jsi#]`Pk#O&501x0pA]#< `+KRs!!rLFf:+6sDdB X S AO?  FMicrosoft Equation 3.0 DS Equation Equation.39qkIvI xtained frmo "from4444554444554444manufacturing two steel products ("W2%x} j e6v`!x} j e6@`!x5O; PA*XSx-;Yx${۝73;PF zRU֮(!Yҧ!C]㿮KSS@.PRS4JWiKǂNuY|$Mғ#4Qc澮&C٦%OϞ%x_t& DdB Y S AO? X2%x} j e6}v`!x FMicrosoft Equation 3.0 DS Equation Equation.39qJJJJJIJJJIxb((L5JJJJJIJJJIx(X L5J JJJJI J J JI,PX L5S@4 and " kIvI xxb((L5J J J J JI J J JIx(X L5 J&J J J JI&J&J&JI,PX L5S@&J&J&JI]xb((L5%J&J&J&J&JI&J} j e6@`!x5O; PA*XSx-;Yx${۝73;PF zRU֮(!Yҧ!C]㿮KSS@.PRS4JWiKǂNuY|$Mғ#4Qc澮&C٦%OϞ%x_t& DdTB Z S AP? Y2Mj ٕW)8v`!!j ٕW\ XJxcdd``^ @bIx(X L5&J(J&J&J&JI(J(J(JI,PX L5S@(J(J(JI]x(XM5"GJJGJGJGJGJJJJGJ,PXM5S@JJJ FMicrosoft Equation 3.0 DS Equation Equation.39qI I ! 1D"L1JE `x0 Yjl R A@2 Nj:P5< %! `fRvF+B2sSRs\ F\22Mhd׊D pM&0pA]v= `f``Ĥ\Y\ )k f22044Dd B [ S AQ? 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2%_YNF1Sjw FMicrosoft Equation 3.0 DS Equation Equation.39q 919 L 2445 919 mt 2447 921 L 2447 921 mt 2450 926 L 2450 926 mt 2451 927 L 2451 927 mt 2453 930 L 2453 930 mt 2456 934 L 2456 934 mt 2456 935 L 2456 935 mt 2458 938 L 2458 938 mt 2461 942 L 2461 942 mt 2462 943 L 2462 943 mt 2464 947 L 2464 947 mt 2467 951 L 2467 951 mt 2467 951 L 2467 951 mt 2469 955 L 2469 955 mt 2472 959 L 2472 959 mt 2472 960 L 2472 960 mt 2475 964 L 247 FMicrosoft Equation 3.0 DS Equation Equation.39q 976 mt 2483 977 L 2483 977 mt 2485 981 L 2485 981 mt 2488 985 L 2488 985 mt 2488 985 L 2488 985 mt 2491 989 L 2491 989 mt 2493 993 L 2493 993 mt 2494 994 L 2494 994 mt 2496 997 L 2496 997 mt 2499 1002 L 2499 1002 mt 2499 1003 L 2499 1003 mt 2501 1006 L 2501 1006 mt 2504 1010 L 2504 1010 mt 2505 1011 L 2505 1011 mt 2506 1014 L 2506 1014 mt 2509 1019 L 2509 1019 mt 2510 1020 L 2510 1020 mt 25kIvI x mt 2515 1029 L 2515 1029 mt 2517 1031 L 2517 1031 mt 2519 1036 L 2519 1036 mt 2521 1038 L 2521 1038 mt 2522 1040 L 2522 1040 mt 2524 1044 L 2524 1044 mt 2526 1047 L 2526 1047 mt 2527 1048 L 2527 1048 mt 2529 1052 L 2529 1052 mt 2531 1056 L 2531 1056 mt 2532 1057 L 2532 1057 mt 2534 1061 L 2534 1061 mt 2537 1065 L 2537 1065 mt 2537 1066 L 2537 1066 mt 2539 1069 L 2539 1069 mt 2541 1074 L 2541 1074 mt 2542 1075 L 2542 1075 mt 2544 1078 L 2544 1078 mt 2546 1082 L 25B { S AO? 2%x} j e6$Xv`!x} j e6@`!x5O; PA*XSx-;Yx${۝73;PF zRU֮(!Yҧ!C]㿮KSS@.PRS4JWiKǂNuY|$Mғ#4Qc澮&C٦%OϞ%x_t& DdH@B  S _YNF1Sjw `Ƚ!x5N; PO>`vz/h($ vVv(^0[ew+Fդ6mP"E˲TdL]EL ԖiXo҈i()uWq|,b 0o>4nd.، ڥI.cڧq&wSa9ۤ>hv״Yv`!60۬&DU0p?/@" xcdd``~ @bD"L1JE `xX,56~) M @ Tv 7$# !L av@Hfnj_jBP~nbC3%@y mĕRl[] `5nAfܤ&`~@2_t3  b#ܤ 8d.hpC uk=pfdbR ,.IͅZf:DlBRDdH@B  S An?(B銗PS$~OK[v`!65xeK}A9VS@" xcdd``~ @bD"L1JE `xX,56~) M @ Tv 7$# !L av@Hfnj_jBP~nbBKt@ڈ+уηNk݂f I9 Ls>d3Hb 2t!$FI-`=p e \P 0y{Ĥ\Y\ 1ٵ`uiB~DdpB  S Ao?'h鄍T"Ay>c IKhQ&f"xlX{tB۩ B$M *r ~Ig{Y5>tnGZNLt=AW ӌs yCgl We"W躋ũ躋)[0_aڛT2"Ҍ!~GnQ~['JZq :™yī''?ջK2 '+x;xM N+NVp7&suBM.lwqqNABJf˩A}]e|4DdphB  S A? 2=gwv'B FMicrosoft Equation 3.0 DS Equation Equation.39qL 2859 1891 mt 2860 1895 L 2860 1895 mt 2861 1899 L 2861 1899 mt 2862 1903 L 2862 1903 mt 2863 1907 L 2863 1907 mt 2863 1912 L 2863 1912 mt 2864 1916 L 2864 1916 mt 2864 1916 L 2864 1916 mt 2865 1920 L 2865 1920 mt 2866 1924 L 2866 1924 mt 2867 1929 L 2867 1929 mt 2868 1933 L 2868 1933 mt 2869 1937 L 2869 1937 mt 2870 1941 L 2870 1941 mt 2870 1942 L 2870 1942 mt 2870 1946 L 2870 1946 mt 2871 1950 L 2871 195 FMicrosoft Equation 3.0 DS Equation Equation.39qmt 2875 1969 L 2875 1969 mt 2875 1971 L 2875 1971 mt 2876 1975 L 2876 1975 mt 2877 1979 L 2877 1979 mt 2878 1984 L 2878 1984 mt 2879 1988 L 2879 1988 mt 2879 1992 L 2879 1992 mt 2880 1996 L 2880 1996 mt 2880 1998 L 2880 1998 mt 2881 2001 L 2881 2001 mt 2882 2005 L 2882 2005 mt 2882 2009 L 2882 2009 mt 2883 2013 L 2883 2013 mt 2884 2017 L 2884 2017 mt 2885 2022 L 2885 2022 mt 2885 2026 L 2885 2026 mt 2886 20kIvI B(x;)=f(x)"log("c i (x)) i=1m I " +12h i2i=1m E " (x),73 L 2893 2073 mt 2894 2077 L 2894 2077 mt 2894 2081 L 2894 2081 mt 2895 2085 L 2895 2085 mt 2896 2089 L 2896 2089 mt 2896 2094 L 2896 2094 mt 2897 2096 L 2897 2096 mt 2897 2098 L 2897 2098 mt 2897 2102 L 2897 21 2i=WI;Yh HQV$b]v`!Zi=WI;Yh HQV$`-P(xkQgfצɦ6m[=HRADGVIm)@Sb! d!uߩ! 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