Arrangement of level sets of quadratic constraints and its relation to nonconvex quadratic optimization problems

TL;DR

This paper investigates a specific class of non-convex quadratic optimization problems with two quadratic constraints, revealing conditions under which strong duality holds without Slater conditions, and explaining the complexity when level sets are arranged alternatively.

Contribution

It introduces a geometric perspective on the arrangement of level sets in quadratic constraints and establishes strong duality results for cases without Slater conditions, extending previous work.

Findings

Strong duality holds for certain non-convex quadratic problems without Slater conditions.

The geometric view explains the difficulty when level sets are arranged alternatively.

Results encompass Ye and Zhang's work and generalized trust region subproblems.

Abstract

We study a special class of non-convex quadratic programs subject to two (possibly indefinite) quadratic constraints when the level sets of the constraint functions are {\it not} arranged {\it alternatively.} It is shown in the paper that this class of problems admit strong duality following a tight SDP relaxation, without assuming primal or dual Slater conditions. Our results cover Ye and Zhang's development in 2003 and the generalized trust region subproblems (GTRS) as special cases. Through the novel geometric view and some simple examples, we can explain why the problem becomes very hard when the level sets of the constraints are indeed arranged alternatively.