Polyak's convexity theorem, Yuan's lemma and S-lemma: extensions and applications

TL;DR

This paper extends key quadratic form theorems, including Polyak's convexity theorem, Yuan's lemma, and the S-lemma, broadening their applicability in quadratic optimization and establishing new duality results.

Contribution

It generalizes Polyak's convexity theorem to any number of quadratic forms and extends Yuan's lemma and S-lemma, enhancing their use in optimization theory.

Findings

Extended Polyak's theorem to multiple quadratic forms with positive definite combinations

Developed a more general Yuan's lemma for second-order optimality conditions

Revealed strong duality in quadratic optimization with bilateral constraints

Abstract

We extend Polyak's theorem on the convexity of joint numerical range from three to any number of quadratic forms on condition that they can be generated by three quadratic forms with a positive definite linear combination. Our new result covers the fundamental Dines's theorem. As applications, we further extend Yuan's lemma and S-lemma, respectively. Our extended Yuan's lemma is used to build a more generalized assumption than that of Haeser (J. Optim. Theory Appl. 174(3): 641-649, 2017), under which the standard second-order necessary optimality condition holds at local minimizer. The extended S-lemma reveals strong duality of homogeneous quadratic optimization problem with two bilateral quadratic constraints.