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

TL;DR

This study develops a hierarchy of semidefinite relaxations based on moment and sum-of-squares representations of the COP and CP cones, and shows that the tightness of the hierarchy is guaranteed under mild conditions.

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.

Figures (2)

• Figure 1: Cyclic allocation graph with $N=5$.
• Figure 2: Visualization of Defender (black straight lines) and Attacker (red curved lines) Strategies in Cyclic Blotto Game with $N=4$ and $N = 5$. Each numbered node represents a location arranged in a circle. Black straight lines connect the pairs of locations chosen in each pure strategy of the defender, representing maximally spaced occupation. Red curved lines connect adjacent location pairs chosen in each pure strategy of the attacker, representing concentrating force and minimize the defender's ability to redeploy support. All strategies are used with equal probability within each player's mixed equilibrium strategy.

Theorems & Definitions (29)

• Lemma 1
• Lemma 2
• Lemma 3
• Proposition 1
• Lemma 4
• Theorem 1
• Proposition 2
• Lemma 5
• Remark
• Remark