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

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.