Tangencies and Polynomial Optimization

Given a polynomial function f colon ℝn rightarrow ℝ and a unbounded basic closed semi-algebraic set S subset ℝn, in this paper we show that the conditions listed below are characterized exactly in terms of the so-called em tangency variety of f on S: (i) The f is bounded from below on S; (ii) The f attains its infimum on S; (iii) The sublevel set x ∈ S | f(x) ≤ λ for λ ∈ ℝ is compact; (iv) The f is coercive on S. Besides, we also provide some stability criteria for boundedness and coercivity of f on S.