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14
Interfaces to PATH 3.0: Design, Implementation and Usage
 Computational Optimization and Applications
, 1998
"... Several new interfaces have recently been developed requiring PATH to solve a mixed complementarity problem. To overcome the necessity of maintaining a different version of PATH for each interface, the code was reorganized using objectoriented design techniques. At the same time, robustness issues ..."
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Cited by 48 (17 self)
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Several new interfaces have recently been developed requiring PATH to solve a mixed complementarity problem. To overcome the necessity of maintaining a different version of PATH for each interface, the code was reorganized using objectoriented design techniques. At the same time, robustness issues were considered and enhancements made to the algorithm. In this paper, we document the external interfaces to the PATH code and describe some of the new utilities using PATH. We then discuss the enhancements made and compare the results obtained from PATH 2.9 to the new version. 1 Introduction The PATH solver [12] for mixed complementarity problems (MCPs) was introduced in 1995 and has since become the standard against which new MCP solvers are compared. However, the main user group for PATH continues to be economists using the MPSGE preprocessor [36]. While developing the new PATH implementation, we had two goals: to make the solver accessible to a broad audience and to improve the effecti...
The Semismooth Algorithm for Large Scale Complementarity Problems
, 1999
"... Complementarity solvers are continually being challenged by modelers demanding improved reliability and scalability. Building upon a strong theoretical background, the semismooth algorithm has the potential to meet both of these requirements. We briefly discuss relevant theory associated with th ..."
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Cited by 19 (7 self)
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Complementarity solvers are continually being challenged by modelers demanding improved reliability and scalability. Building upon a strong theoretical background, the semismooth algorithm has the potential to meet both of these requirements. We briefly discuss relevant theory associated with the algorithm and describe a sophisticated implementation in detail. Particular emphasis is given to robust methods for dealing with singularities in the linear system and to large scale issues. Results on the MCPLIB test suite indicate that the code is robust and has the potential to solve very large problems.
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization
, 2002
"... Constrained optimization has been extensively used to... This paper briefly reviews some methods available to solve these problems and describes a new suite of tools for working with MPEC models. Computational results demonstrating... ..."
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Cited by 13 (3 self)
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Constrained optimization has been extensively used to... This paper briefly reviews some methods available to solve these problems and describes a new suite of tools for working with MPEC models. Computational results demonstrating...
Algorithms and Environments for Complementarity
, 2000
"... Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The ..."
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Cited by 6 (0 self)
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Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The main focus of this thesis is algorithms and environments for solving complementarity problems. Environments, such as AMPL and GAMS, are used by practitioners to easily write large, complex models. Support for these packages is provided by PATH 4.x and SEMI through the customizable solver interface specified in this thesis. The main design feature is the abstraction of core components from the code with implementations tailored to a particular environment supplied either at compile or run time. This solver interface is then used to develop new links to the MATLAB and NEOS tools. Preprocessing techniques are an integral part of linear and mixed integer programming codes and are primarily used to reduce the size and complexity of a model prior to solving it. For example, wasted computation is avoided when an infeasible model is detected.
Preprocessing Complementarity Problems
, 2001
"... Preprocessing techniques are extensively used by the linear and integer programming communities as a means to improve model formulation by reducing size and complexity. Adaptations and extensions of these methods for use within the complementarity framework are detailed. The preprocessor developed i ..."
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Cited by 3 (2 self)
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Preprocessing techniques are extensively used by the linear and integer programming communities as a means to improve model formulation by reducing size and complexity. Adaptations and extensions of these methods for use within the complementarity framework are detailed. The preprocessor developed is comprised of two phases. The rst recasts a complementarity problem as a variational inequality over a polyhedral set and exploits the uncovered structure to x variables and remove constraints. The second discovers information about the function and utilizes complementarity theory to eliminate variables. The methodology is successfully employed to preprocess several models. Keywords: mixed complementarity, preprocessing 1. INTRODUCTION General purpose codes for solving complementarity problems have previously lacked one signicant feature: a powerful preprocessor. The benets of preprocessing have long been known to the linear [1, 2] and integer [19] programming communities, yet have no...
An Extended Mathematical Programming Framework
, 2009
"... Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to ..."
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Cited by 3 (1 self)
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Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.
Case Studies in Complementarity: Improving Model Formulation
 Ill{Posed Variational Problems and Regularization Techniques, number 477 in Lecture Notes in Economics and Mathematical Systems
, 1998
"... Over the past several years, many practitioners have been formulating nonlinear variational inequalities as mixed complementarity problems within modeling languages such as GAMS and AMPL. Sometimes the models generated are poorly specified, either because the function is undefined near the solut ..."
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Cited by 3 (2 self)
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Over the past several years, many practitioners have been formulating nonlinear variational inequalities as mixed complementarity problems within modeling languages such as GAMS and AMPL. Sometimes the models generated are poorly specified, either because the function is undefined near the solution or the problem is illconditioned or singular. In this paper, we look at information provided by the PATH solver about the model that can be used to identify problem areas and improve formulation. Descriptions and uses of the data provided are detailed via several case studies. 1 Introduction Developing a practical model of a complex situation is a difficult task in which an approximate representation is initially constructed and then iteratively refined until an accurate formulation is obtained. During the intermediate stages, the models generated have a tendency to be illdefined, illconditioned, and/or singular. Information generated by a solver can help the modeler to detect t...
Formulating and Solving Nonlinear Programs as Mixed Complementarity Problems
 Optimization. Lecture Notes in Economics and Mathematical Systems
, 2000
"... . We consider a primaldual approach to solve nonlinear programming problems within the AMPL modeling language, via a mixed complementarity formulation. The modeling language supplies the first order and second order derivative information of the Lagrangian function of the nonlinear problem using au ..."
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Cited by 3 (0 self)
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. We consider a primaldual approach to solve nonlinear programming problems within the AMPL modeling language, via a mixed complementarity formulation. The modeling language supplies the first order and second order derivative information of the Lagrangian function of the nonlinear problem using automatic differentiation. The PATH solver finds the solution of the first order conditions which are generated automatically from this derivative information. In addition, the link incorporates the objective function into a new merit function for the PATH solver to improve the capability of the complementarity algorithm for finding optimal solutions of the nonlinear program. We test the new solver on various test suites from the literature and compare with other available nonlinear programming solvers. Keywords: Complementarity problems, nonlinear programs, automatic differentiation, modeling languages. 1 Introduction While the use of the simplex algorithm for linear programs in the 1940's h...
Modeling and Solution Environments for MPEC: GAMS
 Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods
, 1998
"... We describe several new tools for modeling MPEC problems that are built around the introduction of an MPEC model type into the GAMS language. We develop subroutines that allow such models to be communicated directly to MPEC solvers. This library of interface routines, written in the C language, prov ..."
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We describe several new tools for modeling MPEC problems that are built around the introduction of an MPEC model type into the GAMS language. We develop subroutines that allow such models to be communicated directly to MPEC solvers. This library of interface routines, written in the C language, provides algorithmic developers with access to relevant problem data, including for example, function and Jacobian evaluations. A MATLAB interface to the GAMS MPEC model type has been designed using the interface routines. Existing MPEC models from the literature have been written in GAMS, and computational results are given that were obtained using all the tools described. Keywords Complementarity, Algorithm, MPEC, Modeling 1 Introduction The Mathematical Program with Equilibrium Constraints (MPEC) arises when one seeks to optimize an objective function subject to equilibrium contraints. These equilibrium constraints may take the form of a variational inequality or complementarity problem, o...