Characterizing the Convexity of Joint-Range for a Pair of Inhomogeneous Quadratic Functions and Strong Duality

Abstract

We establish various extensions of the convexity Dines theorem for a (joint-range) pair of inhomogeneous quadratic functions. If convexity fails we describe those rays for which the sum of the joint-range and the ray is convex. Afterwards, we derive a characterization of the convexity of the joint-range itself. The convexity Dines theorem for a pair of homogeneous quadratic functions and its extension for inhomogenous functions due to Polyak are re-obtained as consequences. These results are suitable for dealing nonconvex inhomogeneous quadratic optimization problems under one quadratic equality or inequality constraint. As applications of our main results, different sufficient conditions for the validity of S-lemma (a nonstrict version of Finsler's theorem) for inhomogeneous quadratic functions, are presented, as well as a new characterization of strong duality (which is a minimax-type result) under Slater-type condition is established. Keywords. Author Keywords:Dines theorem; hidden convexity; simultaneous diagonalization; quadratic programming; nonstrict version of Finsler's theorem; strong duality. KeyWords Plus:S-LEMMA; RECURRING THEOREM; OPTIMIZATION; FORMS; CONSTRAINTS; EXTENSIONS; EXISTENCE; INTERIOR