Steven P. Dirkse

Mcplib: a collection of nonlinear mixed complementarity problems

The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully. The collection is available by anonymous ftp. Computational results using the PATH solver covering all of these problems are described

A Comparison of Large Scale Mixed Complementarity Problem Solvers

This paper provides a means for comparing various computercodes for solving large scale mixed complementarity problems. Wediscuss inadequacies in how solvers are currently compared, andpresent a testing environment that addresses these inadequacies. Thistesting environment consists of a library of test problems, along withGAMS and MATLAB interfaces that allow these problems to be easilyaccessed. The environment is intended for use as a tool byother researchers to better understand both their algorithms and theirimplementations, and to direct research toward problem classes thatare currently the most challenging. As an initial benchmark, eightdifferent algorithm implementations for large scale mixedcomplementarity problems are briefly described and tested with defaultparameter settings using the new testing environment.

Frontiers in Applied General Equilibrium Modeling: Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization

Constrained optimization has been extensively used to solve many large scale deterministic problems arising in economics, including, for example, square systems of equations and nonlinear programs. A separate set of models have been generated more recently, using complementarity to model various phenomenon, particularly in general equilibria. The unifying framework of mathematical programs with equilibrium constraints (MPEC) has been postulated for problems that combine facets of optimization and complementarity. This paper briefly reviews some methods available to solve these problems and described a new suite of tools for working with MPEC models. Computational results demonstrating the potential of this tool are given that automatically construct and solve a variety of different nonlinear programming reformulations of MPEC problems. This material is based on research partially supported by the National Science Foundation Grant CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, Microsoft Corporation and the Guggenheim Foundation.

An extended mathematical programming framework

Abstract 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.

A pathsearch damped Newton method for computing general equilibria

Computable general equilibrium models and other types of variational inequalities play a key role in computational economics. This paper describes the design and implementation of a pathsearch damped Newton method for solving such problems. Our algorithm improves on the typical Newton method (which generates and solves a sequence of LCPs) in both speed and robustness. The underlying complementarity problem is reformulated as a normal map so that standard algorithmic enhancements of Newtons method for solving nonlinear equations can be easily applied. The solver is implemented as a GAMS subsystem, using an interface library developed for this purpose. Computational results obtained from a number of test problems arising in economics are given.

Modeling and Solution Environments for MPEC: GAMS & MATLAB

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.

Traffic Modeling and Variational Inequalities Using GAMS

We describe how several traffic assignment and design problems can be formulated within the GAMS modeling language using newly developed modeling and interface tools. The fundamental problem is user equilibrium, where multiple drivers compete noncooperatively for the resources of the traffic network. A description of how these models can be written as complementarity problems, variational inequalities, mathematical programs with equilibrium constraints, or stochastic linear programs is given. At least one general purpose solution technique for each model format is briefly outlined. Some observations relating to particular model solutions are drawn.

PAVER 2.0: an open source environment for automated performance analysis of benchmarking data

In this paper we describe PAVER 2.0, an environment (i.e. a process and a suite of tools supporting that process) for the automated performance analysis of benchmarking data. This new environment improves on its predecessor by addressing some of the shortcomings of the original PAVER (Bussieck et al. in Global optimization and constraint satisfaction, lecture notes in computer science, vol 2861, pp 223–238. Springer, Berlin, 2003. doi:10.1007/978-3-540-39901-8_17) and extending its capabilities. The changes serve to further the original goals of PAVER (automation of the visualization and summarization of benchmarking data) while making the environment more accessible for the use of and modification by the entire community of potential users. In particular, we have targeted the end-users of optimization software, as they are best able to make the many subjective choices necessary to produce impactful results when benchmarking optimization software. We illustrate with some sample analyses conducted via PAVER 2.0.

Software Quality Assurance for Mathematical Modeling Systems

With increasing importance placed on standard quality assurance methodologies by large companies and government organizations, many software companies have implemented rigorous quality assurance (QA) processes to ensure that these standards are met. The use of standard QA methodologies cuts maintenance costs, increases reliability, and reduces cycle time for new distributions. Modeling systems differ from most software systems in that a model may fail to solve to optimality without the modeling system being defective. This additional level of complexity requires specific QA activities. To make software quality assurance (SQA) more cost-effective, the focus is on reproducible and automated techniques. In this paper we describe some of the main SQA methodologies as applied to modeling systems. In particular, we focus on configuration management, quality control, and testing as they are handled in the GAMS build framework, emphasizing reproducibility, automation, and an open-source public-domain framework.