Hidden convexity of quadratic systems and its application to quadratic programming

TL;DR

This paper explores the hidden convexity properties of quadratic systems, providing new conditions and proofs that enable more effective solutions to quadratic programming problems.

Contribution

It introduces novel sufficient conditions for convexity in quadratic systems and establishes hidden convexity results for trust-region problems and quadratic inequalities.

Findings

Established new sufficient conditions for convexity of quadratic systems

Proved hidden convexity of trust-region problems with linear constraints

Derived necessary and sufficient optimality and duality conditions for quadratic programming

Abstract

In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math. Program. 147, 171--206, 2014}]. We also provide a complete proof of the hidden convexity of a system of two quadratic functions in [{\it J. Glob. Optim. 56, 1045--1072, 2013}]. Furthermore, necessary and sufficient conditions for the S-lemma concerning systems of quadratic inequalities are investigated. Finally, we derive necessary and sufficient global optimality conditions and strong duality results for quadratic programming.