Robust solutions for a class of quadratic optimization problems without classical convexity assumptions

Abstract

We investigate a class of quadratic robust optimization problems under lower and upper bounds on the constraint, and establish, a robust alternative-type result and a robust S-lemma, provided a generalized convexity assumption and a suitable Slater's condition hold. The robust S-lemma allows us to characterize the robust solutions via first and second order optimality conditions. Relationships with strong duality are also proposed. We base our analysis on Dines' theorem concerning the convexity of images of two quadratic forms, and therefore the homogenization procedure plays a fundamental role. Additionally, we present a novel convex image result suitable for situations where existing results elsewhere are not applicable. This is illustrated by a concrete example.