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

TL;DR

This paper establishes an equivalence between solving semidefinite programs and finding optimal strategies in associated zero-sum semidefinite games, under certain regularity conditions.

Contribution

It extends classical LP-game equivalence results to semidefinite programs, addressing previous limitations and providing new duality and solution bounds.

Findings

Semidefinite programs are equivalent to zero-sum semidefinite games under natural conditions.

The game value acts as a certificate for the existence of strongly optimal solutions.

New bounds for solutions of semidefinite programs are derived.

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.