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

Aloïs Duguet, Tobias Harks, Martin Schmidt, Julian Schwarz

TL;DR

This work tackles the computation of equilibria in generalized Nash games with mixed-integer decisions, where exact equilibria may not exist due to nonconvex data. It introduces a branch-and-cut framework that targets $(\alpha,\beta)$-Nash equilibria by reformulating the search as a minimization of a regret bound $\lambda$ and employing proxy variables $\eta$ and $\xi$ to relax best-response values and costs, with cuts based on intersection-cut theory to exclude non-equilibria. The authors establish existence and termination results for generalized and standard NEPs under linear constraints and convex/concave cost structures, and propose an adaptive single-tree binary-search method to compute best-approximate equilibria efficiently. Numerical experiments on mixed-integer flow games demonstrate practical viability, showing that a substantial share of instances admit small multiplicative approximations and that finite termination is achievable under the stated conditions. The framework provides a principled approach to approximate equilibria in nonconvex strategic settings and offers a path toward scalable computation in applications like infrastructure and network 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.

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

TL;DR

This work tackles the computation of equilibria in generalized Nash games with mixed-integer decisions, where exact equilibria may not exist due to nonconvex data. It introduces a branch-and-cut framework that targets -Nash equilibria by reformulating the search as a minimization of a regret bound and employing proxy variables and to relax best-response values and costs, with cuts based on intersection-cut theory to exclude non-equilibria. The authors establish existence and termination results for generalized and standard NEPs under linear constraints and convex/concave cost structures, and propose an adaptive single-tree binary-search method to compute best-approximate equilibria efficiently. Numerical experiments on mixed-integer flow games demonstrate practical viability, showing that a substantial share of instances admit small multiplicative approximations and that finite termination is achievable under the stated conditions. The framework provides a principled approach to approximate equilibria in nonconvex strategic settings and offers a path toward scalable computation in applications like infrastructure and network 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.

Paper Structure

This paper contains 19 sections, 10 theorems, 32 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. $(\alpha,\beta)$-Nash Equilibria
  4. A B&C Framework
  5. New Cuts
  6. Computing Best-Approximate Equilibria
  7. Numerical Case Study
  8. Related Work
  9. Problem Statement
  10. The Algorithm
  11. Cuts and Finite Termination
  12. Generalized Nash Equilibrium Problems
  13. Standard Nash Equilibrium Problems
  14. Adaptive B&C for Best-Approximate Nash Equilibria
  15. Numerical Results for the NEP Case
  16. ...and 4 more sections

Key Result

Lemma 3.1

A strategy profile $x \in \prod_{j \in N}\mathbb{R}^{k_j+l_j}$ is an $(\alpha,\beta)$-NE if and only if there exists a feasible point $(x,\lambda,\eta,\xi)$ for model:R with $\lambda \leq 0$. In particular, there exists an $(\alpha,\beta)$-NE if and only if the optimal value of model:R is smaller or

Figures (2)

  • Figure 1: Sketch of an intersection cut in the context of $(\alpha,\beta)$-NE in mixed-integer GNEPs
  • Figure 2: Left: Number of instances solved with respect to the computation time for instances solved for the simple binary search method (solid red line), the singleTree+Cuts variant (dashed blue line) and the singleTree variant (long-dashed green line). Right: Number of instances for which an $(\alpha,0$)-NE or better was found using the singleTree+Cuts variant.

Theorems & Definitions (20)

  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 10 more