Solving equilibrium problems using extended mathematical programming

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.

• The paper introduces an EMP framework that naturally translates complex equilibrium formulations into MCPs for efficient computation.
• It outlines specific strategies like replication, switching, and substitution for managing shared variables and constraints.
• Practical implementations in oligopolistic market and general equilibrium models demonstrate the framework's potential to simplify complex economic models.

Extended Mathematical Programming for Equilibrium Problems

In "Solving equilibrium problems using extended mathematical programming", the authors introduce a robust framework that extends traditional mathematical programming to effectively tackle equilibrium problems and their variational representations. This framework is designed to handle a variety of complex equilibrium model formulations, including but not limited to generalized Nash equilibrium problems (GNEPs), multiple optimization problems with equilibrium constraints (MOPECs), and quasi-variational inequalities (QVIs). The implementation is facilitated through modeling languages such as AMPL, GAMS, and Julia.

Framework Overview

The extended mathematical programming (EMP) framework provides a systematic approach to specify equilibrium and variational problems and compute their solutions efficiently. It introduces constructs that allow modelers to annotate variables and equations, facilitating the natural translation of models to computationally tractable forms without requiring manual derivative calculations. This is particularly beneficial in handling sophisticated model structures such as shared constraints and shared variables.

Shared Constraints and Variables

Shared Constraints: In equilibrium models, constraints shared across multiple agents create complex interdependencies. The framework allows these shared constraints to be naturally specified. It facilitates switching between GNEP equilibria and variational equilibria, thus enabling a more comprehensive analysis of equilibrium states.

Shared Variables: The EMP framework introduces the concept of shared variables, which are essential in modeling realistic scenarios like mixed pricing strategies among agents. These are implemented through implicit variables that gather shared values across agents without replicating variable instances, thereby optimizing computational efficiency.

MCP Formulations

To solve equilibrium problems, the framework transforms them into mixed complementarity problems (MCPs), which are compatible with existing solvers like Path. Various MCP formulation strategies are as follows:

• Replication: Each shared variable is duplicated for each agent, typically increasing problem size.
• Switching: Free variables are exchanged with multipliers, minimizing the need for duplication and leading to reduced problem size.
• Substitution: When the implicit function theorem holds, shared variables can be substituted with functions of other variables, reducing MCP size significantly when explicit algebraic expressions are known.

Practical Implementations

The EMP framework is applied to real-world problems such as oligopolistic market equilibria, general equilibrium models with equilibrium constraints (EPECs), and models with mixed pricing agents. These applications demonstrate the framework's flexibility and power in simplifying complex models and enabling efficient solution derivation through its high-level constructs.

Conclusion and Future Work

The paper presents a comprehensive approach to formulating and solving equilibrium problems by leveraging extended mathematical programming. Future developments could enhance the framework to incorporate more specialized problem types, including stochastic variations and more complex hierarchical models, broadening its applicability across various domains of economic modeling and operations research.

Implications

The enhanced EMP framework significantly reduces model complexity, aligns computational representation closely with theoretical formulations, and increasingly empowers modelers to tackle large-scale, complex equilibrium scenarios proficiently. The framework’s integration with modeling languages ensures accessibility and adaptability, promising impactful advancements in economic modeling and decision-making processes.

The work done by the authors lays a solid groundwork for future exploration into decomposition algorithms and further integration into other modeling environments, showcasing the potential to evolve mathematical programming paradigms toward handling increasingly intricate equilibrium challenges.