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.

翻译：根据Dantzig (1951)和Adler (2013)的研究结果，计算线性规划的最优解等价于寻找零和双矩阵博弈中的最优策略。Dantzig的原始结果存在不完整性，原因在于其将线性规划归约为零和博弈的方法并非对所有线性规划都成立。本文证明，在要求存在强最优原对偶解或严格无界方向的自然约束规格下，计算半正定规划的解等价于寻找关联零和半正定博弈中的最优策略。我们的工作建立在Ickstadt、Theobald和Tsigaridas (2024)的研究基础上，其归约方法类似于Dantzig的工作，无法处理某一特定子类的半正定规划。我们证明该等价性结果的主要论证工具包括：(i) von Stengel (2023)对Dantzig构造的扩展在半正定框架下的推广；(ii) 处理半正定情形中更一般的对偶现象的技术；以及(iii) 半正定规划解的坐标的显式界。作为副产品，博弈值提供了认证性：当且仅当强最优解存在时该值为零，否则最优策略将给出原问题或对偶问题的不可行性认证。