On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity

TL;DR

This paper develops new converging hierarchies for polynomial optimization using classical positivity certificates, simplifying the process by relying on global positivity and classical results, with potential for more efficient computational methods.

Contribution

It introduces hierarchies based on early 20th-century positivity certificates, including Artin's and Polyá's results, reducing reliance on complex Positivstellensätze for polynomial optimization.

Findings

Hierarchies based on classical positivity certificates converge for polynomial minimization.

New LP and SOCP hierarchies utilize dsos and sdsos polynomials.

Theoretical framework with potential for more efficient optimization algorithms.

Abstract

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 Positivstellens\"atze from the late 20th century (e.g., due to Stengle, Putinar, or Schm\"udgen) 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 Positivstellens\"atze 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 of Poly\'a on global positivity of even forms to construct an "optimization-free" converging hierarchy for general polynomial minimization problems. This hierarchy only requires polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming and second-order cone programming-based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of dsos and sdsos 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 basic arithmetic at each level-require the use of bisection and increase the number of variables (resp. degree) of the problem by the number of inequality constraints plus three (resp. by a factor of two).