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Let $V$ be a real algebraic variety with singularities and $f$ be a realpolynomial non-negative on $V$. Assume that the regular locus of $V$ is densein $V$ by the usual topology. Using Hironaka's resolution of singularities andDemmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-basedrepresentation that characterizes the non-negativity of $f$ on $V$. Thisrepresentation allows us to build up exact semidefinite relaxations forpolynomial optimization problems whose optimal solutions are possiblysingularities of the constraint sets.

A Nichtnegativstellensatz on singular varieties under the denseness of regular loci