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Best-response algorithms for a class of monotone Nash equilibrium problems with mixed-integer variables

Filippo Fabiani, Simone Sagratella

TL;DR

The paper addresses mixed-integer Nash equilibrium problems (MI-NEPs) that become monotone NEPs upon relaxing integrality. It develops and analyzes Jacobi/Gauss-Seidel best-response algorithms, proving that the BR iterates eventually enter a bounded region containing all MI-NEP solutions, with the region’s size governed by a contraction modulus $\alpha$ and a discrete-bound $\beta$. For the continuous relaxation, the BR scheme converges to the unique NEP solution with a data-dependent complexity bound, and the authors present a rare sufficient condition guaranteeing the existence of MI-NEP solutions, linking them to the continuous relaxation. The results are validated through a smart-building control example, illustrating practical benefits such as initializing from the continuous relaxation and using reduced feasible regions to accelerate computation. Overall, the work provides theoretical guarantees and practical guidance for solving MI-NEPs via BR methods in applications where integrality constraints are natural yet tractable through continuous relaxations.

Abstract

We characterize the convergence properties of traditional best-response (BR) algorithms in computing solutions to mixed-integer Nash equilibrium problems (MI-NEPs) that turn into a class of monotone Nash equilibrium problems (NEPs) once relaxed the integer restrictions. We show that the sequence produced by a Jacobi/Gauss-Seidel BR method always approaches a bounded region containing the entire solution set of the MI-NEP, whose tightness depends on the problem data, and it is related to the degree of strong monotonicity of the relaxed NEP. When the underlying algorithm is applied to the relaxed NEP, we establish data-dependent complexity results characterizing its convergence to the unique solution of the NEP. In addition, we derive one of the very few sufficient conditions for the existence of solutions to MI-NEPs. The theoretical results developed bring important practical benefits, illustrated on a numerical instance of a smart building control application.

Best-response algorithms for a class of monotone Nash equilibrium problems with mixed-integer variables

TL;DR

The paper addresses mixed-integer Nash equilibrium problems (MI-NEPs) that become monotone NEPs upon relaxing integrality. It develops and analyzes Jacobi/Gauss-Seidel best-response algorithms, proving that the BR iterates eventually enter a bounded region containing all MI-NEP solutions, with the region’s size governed by a contraction modulus and a discrete-bound . For the continuous relaxation, the BR scheme converges to the unique NEP solution with a data-dependent complexity bound, and the authors present a rare sufficient condition guaranteeing the existence of MI-NEP solutions, linking them to the continuous relaxation. The results are validated through a smart-building control example, illustrating practical benefits such as initializing from the continuous relaxation and using reduced feasible regions to accelerate computation. Overall, the work provides theoretical guarantees and practical guidance for solving MI-NEPs via BR methods in applications where integrality constraints are natural yet tractable through continuous relaxations.

Abstract

We characterize the convergence properties of traditional best-response (BR) algorithms in computing solutions to mixed-integer Nash equilibrium problems (MI-NEPs) that turn into a class of monotone Nash equilibrium problems (NEPs) once relaxed the integer restrictions. We show that the sequence produced by a Jacobi/Gauss-Seidel BR method always approaches a bounded region containing the entire solution set of the MI-NEP, whose tightness depends on the problem data, and it is related to the degree of strong monotonicity of the relaxed NEP. When the underlying algorithm is applied to the relaxed NEP, we establish data-dependent complexity results characterizing its convergence to the unique solution of the NEP. In addition, we derive one of the very few sufficient conditions for the existence of solutions to MI-NEPs. The theoretical results developed bring important practical benefits, illustrated on a numerical instance of a smart building control application.
Paper Structure (13 sections, 11 theorems, 43 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 11 theorems, 43 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem definition and main assumptions
  3. Best-response methods
  4. About continuous NEPs
  5. Relations between MI-NEPs and their continuous NEP relaxations, and a discussion about existence of solutions
  6. Remark about inexact BR algorithms
  7. Discussion on Assumptions \ref{['as: block-contraction']} and \ref{['as: discrete bound']}
  8. Conditions for Assumption \ref{['as: block-contraction']} and their relations with strong monotonicity
  9. On Assumption \ref{['as: discrete bound']}
  10. Practical usage of the BR algorithms and numerical results
  11. Problem description: Local smart building control
  12. Numerical results
  13. Conclusion

Key Result

Theorem 1

Suppose that Assumptions as: block-contraction and as: discrete bound hold true, and that In addition, assume that, in Algorithm alg: Jacobi, every $h$ iterations at least one of any player $\nu$ is computed, that is, $\nu \in \cup_{t = k}^{k + h} {\cal J}^t$ for each player $\nu$ and each iterate $k$. Let $\{{\bf x}^k\}_{k \in \mathbb{N}} \subseteq \Omega$ be the sequence generated by A $\square

Figures (2)

  • Figure 1: Mean value (solid blue line) and standard deviation (shaded blue area) of the players' worst-case distance from the continuous equilibrium of the relaxed , computed in those numerical instances for which Algorithm \ref{['alg: Jacobi']} has not converged for $^\text{u}_\text{f}$. The red line denotes, instead, the bound in \ref{['eq: error bound wrt continuous NEP']} with $\gamma=1.001$ (note the different values on the left-right ordinates).
  • Figure 2: Example of equilibria computed in all the considered cases. Specifically, (a) Relaxed ; with granularity up to units over the entire feasible set (b) and over the reduced one (c); with granularity up to tens over the entire feasible set (d) and over the reduced one (e).

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Example 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Example 2
  • Example 3
  • Proposition 2
  • ...and 8 more