On the equivalence of semidefinite programming and zero-sum semidefinite games

2026-04-24

AI Summary

This work reformulates the problem of solving semidefinite programs (SDPs) as the search for optimal strategies in zero-sum semidefinite games, particularly targeting instances that challenge existing methods. Under natural constraint qualifications, it establishes a constructive and complete equivalence between primal-dual SDPs and zero-sum semidefinite games: the game value is zero if and only if a strong optimal solution exists; otherwise, the framework yields an infeasibility certificate for either the primal or the dual problem. By integrating SDP duality theory, a semidefinite generalization of von Stengel’s construction, techniques for handling generalized duality phenomena, and explicit bounds on solution coordinates, this approach extends applicability to a broader class of SDPs and overcomes limitations of prior methods.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

By results of Dantzig (1951) and Adler (2013), computing the optimal solutions of a linear program is equivalent to finding optimal strategies in zero-sum bimatrix games. Dantzig's original result was incomplete, in the sense that the reduction of a linear program to a zero-sum game did not work for all possible linear programs. We show that, under a natural constraint qualification requiring either the existence of strongly optimal primal-dual solutions or of a strictly unbounded direction, computing the solution of a semidefinite program is equivalent to finding optimal strategies in an associated zero-sum semidefinite game. Our work builds upon Ickstadt, Theobald, and Tsigaridas (2024), where, similar to Dantzig's work, the proposed reduction cannot handle a certain subclass of semidefinite programs. Our main proof ingredients for the equivalence result include: (i) a semidefinite generalization of von Stengel's (2023) extension of Dantzig's construction; (ii) techniques for handling more general duality phenomena in the semidefinite setting; and (iii) an explicit bound for the (coordinates) of the solutions of a semidefinite program. As a by-product, the game value provides a certificate: it is zero if and only if strongly optimal solutions exist, and otherwise optimal strategies yield an infeasibility certificate for the primal or dual program.

Problem

• semidefinite programming
• zero-sum semidefinite games
• equivalence
• duality
• constraint qualification

Innovation

• semidefinite programming
• zero-sum semidefinite games
• strong duality
• infeasibility certificate
• constraint qualification