On globally solving nonconvex trust region subproblem via projected gradient method

TL;DR

This paper demonstrates that the projected gradient method almost always converges to the global minimum when applied to a reformulation of the trust region subproblem, offering a new approach for solving nonconvex quadratic optimization problems.

Contribution

It reveals that a simple projected gradient method can globally solve nonconvex trust region subproblems with linear convergence in the easy case, regardless of initial point.

Findings

Projected gradient converges to global minimizer in most cases.

Linear local convergence rate in the easy case.

Applicable to equality-constrained trust region subproblems.

Abstract

The trust region subproblem (TRS) is to minimize a possibly nonconvex quadratic function over a Euclidean ball. There are typically two cases for (TRS), the so-called ``easy case'' and ``hard case''. Even in the ``easy case'', the sequence generated by the classical projected gradient method (PG) may converge to a saddle point at a sublinear local rate, when the initial point is arbitrarily selected from a nonzero measure feasible set. To our surprise, when applying (PG) to solve a cheap and possibly nonconvex reformulation of (TRS), the generated sequence initialized with {\it any} feasible point almost always converges to its global minimizer. The local convergence rate is at least linear for the ``easy case'', without assuming that we have possessed the information that the ``easy case'' holds. We also consider how to use (PG) to globally solve equality-constrained (TRS).