Hölderian Error Bounds and Kurdyka-Łojasiewicz 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 Hölderian error bound holds globally for the TRS with modulus 1/4, and the Kurdyka-Łojasiewicz (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 Hölderian error bound modulus and the KL exponent is 1/2. Finally, as a byproduct, we further apply the obtained KL property to show that projected gradient methods studied elsewhere for solving the TRS achieve a local sublinear or even linear rate of convergence with probability 1 by choosing a proper initial point.