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

TL;DR

This paper introduces a branch-and-cut algorithm to compute approximate equilibria in mixed-integer generalized Nash games, addressing the challenge of non-existence of exact equilibria and computational complexity.

Contribution

It develops a novel branch-and-cut method with intersection cuts for mixed-integer games and introduces a single-tree binary-search approach for best-approximate equilibria.

Findings

The method can compute approximate equilibria or prove their non-existence.

The approach terminates finitely under certain conditions for standard Nash problems.

Numerical results demonstrate effectiveness on mixed-integer flow games.

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.