Generalized SOS-Convexity and Strong Duality with SDP Dual Programs in Polynomial Optimization

We introduce the notion of ρ-SOS-convexity, extending the numerically checkable concept of SOS-convexity of a real polynomial. The class of ρ-SOS-convex polynomials includes the important class of (not necessarily convex) quadratic functions. We provide various characterizations of ρ-SOS-convexity in terms of SOS-convexity. Consequently, we establish strong duality results for classes of nonconvex polynomial optimization problems involving strong SOS-convex (where ρ > 0) and weak SOS-convex (where ρ < 0) polynomials. These classes of problems include some polynomial optimization problems, involving SOS-convex polynomials, minimax quadratic optimization problems with quadratic constraints, fractional programming problems and robust optimization problems. Our results also provide necessary and sufficient conditions for strong duality of some classes of minimax quadratic optimization problems and extended trust-region problems.