Finding global solutions for a class of possibly nonconvex QCQP problems through the S-lemma

TL;DR

This paper establishes necessary and sufficient conditions for global optimality in a specific class of nonconvex QCQP problems using a generalized S-Lemma, and proves the exactness of SDP and SOCP relaxations.

Contribution

It introduces a new class of QCQP problems (S-QCQP) and provides a generalized S-Lemma to determine global solutions, including relaxation exactness.

Findings

Necessary and sufficient KKT conditions for S-QCQP

Exactness of SDP relaxation for S-QCQP

Exactness of SOCP relaxation for S-QCQP

Abstract

In this paper we provide necessary and sufficient (KKT) conditions for global optimality for a new class of possibly nonconvex quadratically constrained quadratic programming (QCQP) problems, denoted by S-QCQP. The class consists of QCQP problems where the matrices of the quadratic components are formed by a scalar times the identity matrix. Our result relies on a generalized version of the S-Lemma, stated in the context of general QCQP problems. Moreover, we prove the exactness of the SDP and the SOCP relaxations for S-QCQP.