Generalized SOS-convexity and strong duality with SDP dual programs in polynomial optimization (Q2789232)

The notion of \(\rho\)-SOS-convexity is introduced, in order to extend the numerically checkable concept of SOS-convexity of a real polynomial, motivated by the class of (not necessarily convex) quadratic functions. Various characterizations of the \(\rho\)-SOS-convexity are given in terms of SOS-convexity and they are employed for establishing strong duality results for classes of nonconvex polynomial optimization problems involving strong and SOS-convex polynomials, respectively. These classes of problems include some useful polynomial optimization problems, involving SOS-convex polynomials, minimax quadratic optimization problems with quadratic constraints, fractional programming problems and robust optimization problems. As byproducts, necessary and sufficient conditions for strong duality of some classes of minimax quadratic optimization problems and extended trust-region problems are provided, too, and the paper ends with some hints towards future research in this direction.