Trust--Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization

TL;DR

This paper extends classical trust-region problem properties to cases with linear constraints, establishing exact SDP relaxations and global optimality conditions under a new dimension condition, enabling polynomial-time solutions for robust optimization problems.

Contribution

It proves that the extended trust-region problem with linear constraints retains exact SDP relaxation and strong duality under a new dimension condition, broadening applicability.

Findings

Exact SDP relaxation holds without Slater qualification.

Dimension condition ensures global optimality via Lagrange multipliers.

Robust least squares and SOCP problems can be solved via SDP under mild assumptions.

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 trust-region 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. 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 SDP and so, their solutions can be validated in polynomial time.