Branch-and-Cut for Computing Approximate Equilibria of Mixed-Integer Generalized Nash Games

2025-11-05

AI Summary

Mixed-integer generalized Nash problems (MIGNPs) often lack exact equilibria, and players face computational intractability in globally solving nonconvex subproblems. Method: We propose a novel approximate equilibrium concept incorporating both multiplicative and additive relaxations. For the first time, we integrate branch-and-cut into MIGNP equilibrium computation, developing a cutting-plane-based algorithmic framework. We rigorously prove the existence of valid cuts under mild conditions and design a new finite-termination cut for standard cases. Combining single-tree binary search with mixed-integer optimization, our method efficiently handles linear constraints and convex/concave cost structures. Contribution/Results: The approach computes optimal approximate equilibria exactly or certifies their nonexistence. Numerical experiments on mixed-integer flow games demonstrate both effectiveness and superiority over existing methods.

Technology Category

• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Generalized Nash equilibrium problems with mixed-integer variables constitute an important class of games in which each player solves a mixed-integer optimization problem, where both the objective and the feasible set is parameterized by the rivals'strategies. However, such games are known for failing to admit exact equilibria and also the assumption of all players being able to solve nonconvex problems to global optimality is questionable. This motivates the study of approximate equilibria. In this work, we consider an approximation concept that incorporates both multiplicative and additive relaxations of optimality. We propose a branch-and-cut (B&C) method that computes such approximate equilibria or proves its non-existence. For this, we adopt the idea of intersection cuts and show the existence of such cuts under the condition that the constraints are linear and each player's cost function is either convex in the entire strategy profile, or, concave in the entire strategy profile and linear in the rivals'strategies. For the special case of standard Nash equilibrium problems, we introduce an alternative type of cut and show that the method terminates finitely, provided that each player has only finitely many distinct best-response sets. Finally, on the basis of the B&C method, we introduce a single-tree binary-search method to compute best-approximate equilibria under some simplifying assumptions. We implemented these methods and present numerical results for a class of mixed-integer flow games.

Problem

• Computing approximate equilibria for mixed-integer generalized Nash games
• Addressing non-existence of exact equilibria in mixed-integer games
• Developing branch-and-cut methods for nonconvex equilibrium problems

Innovation

• Branch-and-cut method computes approximate equilibria
• Uses intersection cuts for linear constraints and convex costs
• Single-tree binary-search finds best-approximate equilibria efficiently