Tangencies and polynomial optimization

Abstract

Given a polynomial function $$f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ and an unbounded closed semi-algebraic set $$S \subset {\mathbb {R}}^n,$$ we show that the conditions listed below are characterized exactly in terms of the so-called tangency variety of the restriction of f on S: Besides, we also provide some stability criteria for boundedness and coercivity of f on S.