Solving equilibrium problems using extended mathematical programming

TL;DR

This paper presents an extended mathematical programming framework that simplifies modeling and solving complex equilibrium and variational problems, enabling natural translation between formulations and supporting shared constraints.

Contribution

It introduces new constructs for modeling equilibrium problems, including shared variables, and demonstrates their application and computational benefits within GAMS/EMP.

Findings

Enhanced modeling flexibility for equilibrium problems.

Improved computational performance with shared variables.

Framework implementation in GAMS/EMP available for use.

Abstract

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define a new concept, shared variables, and demonstrate its uses for sparse reformulation, equilibrium problems with equilibrium constraints, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are given. Our framework has been implemented and is available within GAMS/EMP.