Optimizing a linear function over a noncompact real algebraic variety

Abstract

Our aim is to compute such a polynomial Φ of the least possible degree. In [3, 4], Rostalski and Sturmfels explored dualities and their interconnections in the context of polynomial optimization (1.1). Assuming that the feasible regionX is irreducible, compact and smooth, they showed that the optimal value function Φ is represented by the defining equation of the hypersurface dual to the projective closure of X [4, Theorem 5.23]. In the present paper, we prove this conclusion is still true for a noncompact real algebraic variety X, when X is irreducible, smooth and the recession cone of the closure of the convex hull co (X) of X is pointed.