On global solvability of a class of possibly nonconvex QCQP problems in Hilbert spaces

TL;DR

This paper establishes conditions under which KKT conditions guarantee global optimality for nonconvex QCQP problems in Hilbert spaces, using a generalized S-Lemma and convexity of an associated set.

Contribution

It introduces a new set of conditions linking convexity and KKT optimality for nonconvex QCQPs in infinite-dimensional spaces, extending classical finite-dimensional results.

Findings

KKT conditions characterize global optimality under convexity assumptions.

A generalized S-Lemma in Hilbert spaces is proved.

Application to QCQPs with special quadratic constraint forms.

Abstract

We provide conditions ensuring that the KKT-type conditions characterizes the global optimality for quadratically constrained (possibly nonconvex) quadratic programming QCQP problems in Hilbert spaces. The key property is the convexity of a image-type set related to the functions appearing in the formulation of the problem. The proof of the main result relies on a generalized version of the (Jakubovich) S-Lemma in Hilbert spaces. As an application, we consider the class of QCQP problems with a special form of the quadratic terms of the constraints.