Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints

Abstract

Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.

Centered solutions for uncertain linear equations

“…The tractability of Problem (7) strongly relies on the choice of the uncertainty set U. For deriving tractable counterpart of (7) with ellipsoidal uncertainties in A and/or b, we would like to refer to Ghaoui and Lebret (1997), Beck and Eldar (2006) and Jeyakumar and Li (2014). In the following example, we solve Problem (25) for the interval linear system in Example 3.…”

“…Ben-Tal et al (2009) show that Problem (7) under independent interval uncertainties can be reformulated into an SOCP problem. In Ghaoui and Lebret (1997), Beck and Eldar (2006) and Jeyakumar and Li (2014), authors derive an SOCP or a semidefinite programming (SDP) reformulation of Problem (7) under ellipsoidal uncertainties. Burer (2012) and Juditsky and Polyak (2012) solve (7) to find the robust rating vectors for Colley's Matrix Ranking and Google's PageRank, respectively.…”

SINR balancing in the downlink of cognitive radio networks with imperfect channel knowledge

“…Recently, P 1 has been solved in [2] by using the powerful S-procedure [3]. Their result for the case of spherical channel uncertainty can be written as:…”

“…This extension assumed that errors in the channel state information (CSI) are bounded by ellipsoids. Based on this channel matrix uncertainty model, the problem was formulated as a quasi-convex optimization program using the S-procedure [3]. To be precise, in the case of channel uncertainty, the authors first derived an equivalent problem that involved rank-1 constraints on the positive semidefinite (PSD) matrices modeling the beamforming vectors.…”

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

“…Model problems of this form arise from robust B Immanuel M. Bomze immanuel.bomze@univie.ac.at 1 ISOR and VCOR, University of Vienna, Vienna, Austria 2 School of Mathematics and Statistics, University of New South Wales, Sydney, Australia optimization problems under matrix norm or polyhedral data uncertainty [5,20] and the application of the trust region method [15] for solving constrained optimization problems, such as nonlinear optimization problems with nonlinear and linear inequality constraints [9,34]. It covers many important and challenging quadratic optimization (QP) problems such as those with box constraints; trust region problems with additional linear constraints; and the CDT (Celis-Dennis-Tapia or two-ball trust-region) problem [1,9,14,28,38].…”