Mixed Complementarity Programming with GAMS

GAMS (General Algebraic Modelling System) is a computer language which was originally developed to assist economists at the World Bank in the quantitative analysis of economic policy questions (Meeraus (1983), Brooke, Kendrick and Meeraus (1987)). At the time of its conception and initial development, linear programming (LP) was the standard framework in which market equilibrium models were constructed. Subsequently, solution algorithms and codes for general nonlinear programming representations became more common and alternative economic paradigms were developed. 1 During this time, GAMS was extended from an LP matrix-generator to accommodate general nonlinear functions. GAMS was awarded the Computer Science Technical Section prize by the Operations Research Society of America in 1987. The release of a PC-based version of GAMS in 1988 lead to its wide-spread recognition as an effective tool for the development of general purpose large-scale mathematical programming models. This paper introduces new features of the GAMS modelling language which permit formulating economic equilibrium models as systems of nonlinear equations, complementarity problems or variational inequalities. These extensions accommodate market and game-theoretic equilibrium models which are not easily studied in an optimization context. These formulations are here referred to as “mixed complementarity problems” (MCPs), reflecting the fact that they may incorporate mixtures of equations and inequalities. 2 The MCP is a natural format for expressing a variety of economic models for both markets and games. Computational evidence suggests that algorithms for solving MCPs are relatively reliable and efficient, particulary for models which are not natural optimization problems. The development of the GAMS/MCP modelling format was motivated by recent theoretical and practical developments in algorithms for nonlinear complementarity problems and variational inequalities. Some of these algorithms are extensions of classical methods for nonlinear equations. Others exploit the variational structure which is characteristic of many economic equilibrium models (see Dafermos (1983)). The most recent techniques are based on ideas from interior-point algorithms for linear programming (Kojima, Megiddo, Noma and Yoshise (1991)). A survey of developments in the theory and application of these methods is provided by Harker and Pang (1990). At this time, two large-scale solvers are available through GAMS/MCP. MILES (a Mixed Inequality and nonLinear Equation Solver) employs a modified Newton algorithm which was originally developed for applied general equilibrium modelling (see Rutherford (1987, 1993), and Anstreicher, Lee and Rutherford (1992)). PATH is a recently developed solver based on a promising new path-following procedure (Dirkse and Ferris (1993), Ralph (1994)). Both codes incorporate implicit