Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

Abstract

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trustregion problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust (LSP) or (SOCP) under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time. * The authors are grateful to the referees for their valuable suggestions and helpful comments which have contributed to the final preparation of the paper.

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.…”

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

“…The CDT problem, in general, is much more challenging than the well-studied trust region problems and has received much attention lately, see for example [1,5,6,9,14]. The problem (P CDT ) arises from robust optimization [20] as well as applying trust region techniques for solving nonlinear optimization problems with both nonlinear and linear constraints: see [34] for the case of trust region problems with additional linear inequalities and see [6,9] for the case of CDT problems. We will establish simple conditions ensuring exactness of the copositive relaxations and the usual Lagrangian relaxations of the extended CDT problem.…”

“…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].…”

“…Moreover, its solution can be found by solving a dual Lagrangian system or, equivalently, a semidefinite optimization problem (SDP). Unfortunately, these nice features do not continue to hold for the more general extended trust-region problem (P); see [20]. In fact, it has been shown that exactness of Lagrangian (or SDP) relaxation can fail for the CDT problem, or for the trust region problem with only one additional linear inequality constraint.…”