The hierarchy of local minimums in polynomial optimization

• 成果类型: Article
• 署名作者: Nie, Jiawang
• 署名单位: University of California System; University of California San Diego
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0845-2
• 发表日期: 2015
• 页码: 555-583
• 关键词: global optimization; squares; sums; tensor; eigenvalues; relaxations; moments

摘要

This paper studies the hierarchy of local minimums of a polynomial in the vector space . For this purpose, we first compute -minimums, for which the first and second order necessary optimality conditions are satisfied. To compute each -minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums.