Homogenization for polynomial optimization with unbounded sets

• 成果类型: Article
• 署名作者: Huang, Lei; Nie, Jiawang; Yuan, Ya-Xiang
• 署名单位: Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; University of California System; University of California San Diego
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01878-5
• 发表日期: 2023
• 页码: 105-145
• 关键词: jacobian sdp relaxation; positive polynomials; global optimization; moment problems; sums; squares; hierarchy; REPRESENTATIONS; approximation; matrices

摘要

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.