H\"olderian error bounds and Kurdyka-{\L}ojasiewicz inequality for the trust region subproblem

TL;DR

This paper analyzes the geometric properties of the trust region subproblem, establishing error bounds and KL inequalities that lead to convergence rate results for projected gradient methods.

Contribution

It proves global H"olderian error bounds and local KL inequalities for the TRS, providing new insights into its geometric structure and convergence analysis.

Findings

H"olderian error bound with modulus 1/4 for TRS

KL inequality with exponent 3/4 at optimal solutions

Projected gradient methods achieve sublinear or linear convergence rates

Abstract

In this paper, we study the local variational geometry of the optimal solution set of the trust region subproblem (TRS), which minimizes a general, possibly nonconvex, quadratic function over the unit ball. Specifically, we demonstrate that a H\"olderian error bound holds globally for the TRS with modulus 1/4 and the Kurdyka-{\L}ojasiewicz (KL) inequality holds locally for the TRS with a KL exponent 3/4 at any optimal solution. We further prove that unless in a special case, the H\"olderian error bound modulus, as well as the KL exponent, is 1/2. Finally, based on the obtained KL property, we further show that the projected gradient methods studied in [A. Beck and Y. Vaisbourd, SIAM J. Optim., 28 (2018), pp. 1951--1967] for solving the TRS achieve a sublinear or even linear rate of convergence.