Expressing Complementarity Problems in an Algebraic Modeling Language and Communicating Them to Solvers

Diverse problems in optimization, engineering, and exonomics have natural formulations in terms of complementarity conditions, which state (in their simplest form) that either a certain non-negative variable must be zero or a corresponding inequality must hold with equality, or both. A variety of algorithms have been devised for solving problems expressed in terms of complementarity conditions. It is thus attractive to consider extending algebraic modeling languages, which are widely used for sending ordinary equations and inequality constrains to solvers, so that they can express complementarity problems directly. We describe an extension to the AMPL modeling language that can express the most common complementarity conditions in a concise and flexible way, through the introduction of a single new "complements" operator. We present details of an efficient implementation that incorporates an augmented presolve phase to simplify complementarity problems, and that converts complementarity conditions to a canonical form convenient to solver

Expressing Complementarity Problems In An Algebraic Modeling Language And Communicating Them To Solvers

. Diverse problems in optimization, engineering, and economics have natural formulations in terms of complementarity conditions, which state (in their simplest form) that either a certain nonnegative variable must be zero or a corresponding inequality must hold with equality, or both. A variety of algorithms have been devised for solving problems expressed in terms of complementarity conditions. It is thus attractive to consider extending algebraic modeling languages, which are widely used for sending ordinary equations and inequality constraints to solvers, so that they can express complementarity problems directly. We describe an extension to the AMPL modeling language that can express the most common complementarity conditions in a concise and flexible way, through the introduction of a single new "complements" operator. We present details of an e#cient implementation that incorporates an augmented presolve phase to simplify complementarity problems, and that converts complementarit..

Preprocessing Complementarity Problems

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..