On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity

• 成果类型: Article
• 署名作者: Ahmadi, Amir Ali; Hall, Georgina
• 署名单位: INSEAD Business School
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0962
• 发表日期: 2019
• 页码: 1192-1207
• 关键词: Complexity

摘要

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensatze from the late 20th century (e.g., due to Stengle, Putinar, or Schmudgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensatze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming-based hierarchy that relies solely on Artin's solution to Hilbert's 17th problem. We also use a classical result from Polya on global positivity of even forms to construct an optimization-free converging hierarchy for general polynomial minimization problems with compact feasible sets. This hierarchy requires only polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming-based and second-order cone programming-based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of diagonally dominant sum of squares and scaled diagonally dominant sum of squares polynomials. We remark that the scope of this paper is theoretical at this stage, as our hierarchies-though they involve at most two sum of squares constraints or only elementary arithmetic at each level-require the use of bisection and increase the number of variables (respectively, the degree) of the problem by the number of inequality constraints plus three (respectively, by a factor of two).