Strong Duality for Generalized Trust Region Subproblem: S-Lemma with Interval Bounds

TL;DR

This paper proves strong duality for a generalized trust region subproblem with interval bounds using a new S-lemma, resolving an open problem in optimization theory.

Contribution

It introduces the S-lemma with interval bounds and demonstrates its application to establish strong duality under mild conditions.

Findings

Strong duality holds for the interval bounded generalized trust region subproblem.

The S-lemma with interval bounds is a key tool in the proof.

The results answer an open problem posed by Pong and Wolkowicz.

Abstract

With the help of the newly developed S-lemma with interval bounds, we show that strong duality holds for the interval bounded generalized trust region subproblem under some mild assumptions, which answers an open problem raised by Pong and Wolkowicz [Comput. Optim. Appl. 58(2), 273-322, 2014].