Max-Min Bilinear Completely Positive Programs: A Semidefinite Relaxation with Tightness Guarantees

TL;DR

This paper introduces a semidefinite relaxation framework with tightness guarantees for max-min bilinear completely positive programs, enabling exact solutions to complex game-theoretic and robust optimization problems.

Contribution

It develops a hierarchy of semidefinite relaxations for CP cone-based max-min problems, with conditions ensuring tightness and exact solutions in practical applications.

Findings

Semidefinite relaxation achieves exact solutions in cyclic Colonel Blotto game.

Hierarchy of relaxations with tightness guarantees under mild conditions.

Framework extends to distributionally robust optimization and polynomial games.

Abstract

Max-min bilinear optimization models, where one agent maximizes and an adversary minimizes a common bilinear objective, serve as canonical saddle-point formulations in optimization theory. They capture, among others, two-player zero-sum games, robust and distributionally robust optimization, and adversarial machine learning. This study focuses on the subclass whose variables lie in the completely positive (CP) cone, capturing a broad family of mixed-binary quadratic max-min problems through the modelling power of completely positive programming. We show that such problems admit an equivalent single-stage linear reformulation over the COP-CP cone, defined as the Cartesian product of the copositive (COP) and CP cones. Because testing membership in COP cones is co-NP-complete, the resulting COP-CP program inherits NP-hardness. To address this challenge, we develop a hierarchy of semidefinite relaxations based on moment and sum-of-squares representations of the COP and CP cones, and flat truncation conditions are applied to certify the tightness. We show that the tightness of the hierarchy is guaranteed under mild conditions. The framework extends existing CP/COP approaches for distributionally robust optimization and polynomial games. We apply the framework to the cyclic Colonel Blotto game, an extension of Borel's classic allocation contest. Across multiple instances, the semidefinite relaxation meets the flat-truncation conditions and solves the exact mixed-strategy equilibrium.