A characterization for tightness of the sparse Moment-SOS hierarchy

• 成果类型: Article; Early Access
• 署名作者: Nie, Jiawang; Qu, Zheng; Tang, Xindong; Zhang, Linghao
• 署名单位: University of California System; University of California San Diego; Hong Kong Polytechnic University; Hong Kong Baptist University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02223-2
• 发表日期: 2025
• 关键词: polynomial optimization problems; sdp-relaxations; CONVERGENCE; tssos

摘要

This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.