On duality gap with polynomial multipliers for polynomial optimization problems

Abstract

In this paper, we examine a zero-duality gap with polynomial multipliers between a non-convex polynomial program and its associated generalized Lagrangian dual problem. To this end, we introduce a sum of squares certificate of non-negativity, abbreviated as (SC), that allows us to verify the non-negativity of a polynomial over a feasible set of a system of given polynomials. With the help of (SC), we show that the zero-duality gap with polynomial multipliers coincides with the convergent Lasserre hierarchy of semidefinite programming relaxations. This zero-duality gap with polynomial multipliers reduces to the classical zero-duality gap with constant multipliers when considering it in the setting of SOS-convex polynomial programs or quadratic programs with the hidden convexity. We also show that, in the framework of truncated polynomial systems, the zero-duality gap with polynomial multipliers can be checked via the classical Karush–Kuhn–Tucker condition under an additional convexity condition.