A Nichtnegativstellensatz on singular varieties under the denseness of regular loci

TL;DR

This paper develops a sum of squares representation for non-negative polynomials on singular real algebraic varieties with dense regular loci, enabling exact semidefinite relaxations for polynomial optimization problems involving singularities.

Contribution

It introduces a Nichtnegativstellensatz for singular varieties using resolution of singularities, extending sum of squares techniques to broader classes of polynomial optimization problems.

Findings

Provides a sum of squares-based characterization of non-negativity on singular varieties.

Enables exact semidefinite relaxations for optimization problems with singular constraints.

Utilizes resolution of singularities and existing Nichtnegativstellensatz to handle singular cases.

Abstract

Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and Demmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-based representation that characterizes the non-negativity of $f$ on $V$. This representation allows us to build up exact semidefinite relaxations for polynomial optimization problems whose optimal solutions are possibly singularities of the constraint sets.