Uniform Quadratic Optimization and Extensions

TL;DR

This paper investigates uniform quadratic optimization problems, establishing new duality conditions, extending results to general QCQPs, and proposing an improved SOCP-based approximation algorithm with applications to Chebyshev center problems.

Contribution

It introduces a new sufficient condition for strong duality in UQ, extends results to QCQP, and develops a dimension-independent approximation algorithm.

Findings

Established a new sufficient condition for strong duality in UQ.

Extended duality results to general QCQP.

Proposed an improved SOCP-based approximation algorithm with dimension-independent bounds.

Abstract

The uniform quadratic optimizatin problem (UQ) is a nonconvex quadratic constrained quadratic programming (QCQP) sharing the same Hessian matrix. Based on the second-order cone programming (SOCP) relaxation, we establish a new sufficient condition to guarantee strong duality for (UQ) and then extend it to (QCQP), which not only covers several well-known results in literature but also partially gives answers to a few open questions. For convex constrained nonconvex (UQ), we propose an improved approximation algorithm based on (SOCP). Our approximation bound is dimensional independent. As an application, we establish the first approximation bound for the problem of finding the Chebyshev center of the intersection of several balls.