A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants

TL;DR

This paper introduces a second-order cone based method to exactly convexify the trust-region subproblem with conic constraints, enabling efficient solutions and revealing connections to convex quadratic minimization.

Contribution

It provides an exact convex reformulation of the nonconvex TRS under a structural condition, extending previous results and facilitating the use of fast convex optimization methods.

Findings

Exact convex reformulation of TRS under certain conditions

Connection established between nonconvex TRS and convex quadratic minimization

Low-complexity characterization of the convex hull of the epigraph

Abstract

We study the trust-region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the classical TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone (SOC) based approach to derive an exact convex reformulation of the TRS under a structural condition on the conic constraint. Our structural condition is immediately satisfied when there is no additional conic constraints, and it generalizes several such conditions studied in the literature. As a result, our study highlights an explicit connection between the classical nonconvex TRS and smooth convex quadratic minimization, which allows for the application of cheap iterative methods such as Nesterov's accelerated gradient descent, to the TRS. Furthermore, under slightly stronger conditions, we give a low-complexity characterization of the convex hull of the epigraph of the nonconvex quadratic function intersected with the constraints defining the domain without any additional variables. We also explore the inclusion of additional hollow constraints to the domain of the TRS, and convexification of the associated epigraph.