A Bundle Trust Region Algorithm for Minimizing Locally Lipschitz Functions

Abstract

We propose a new algorithm for minimizing locally Lipschitz functions that combines both the bundle and trust region techniques. Based on the bundle methods the objective function is approximated by a piecewise linear working model which is updated by adding cutting planes at unsuccessful trial steps. The algorithm defines, at each iteration, a new trial point by solving a subproblem that employs the working model in the objective function subject to a region, which is called the trust region. The algorithm is studied from both theoretical and practical points of view. Under a assumption on the objective function, global convergence of it is verified to stationary points. In order to demonstrate the reliability and efficiency of the proposed algorithm, a MATLAB implementation of it is prepared and numerical experiments have been made using some academic nonsmooth test problems. Computational results show that the developed method is efficient for solving nonsmooth and nonconvex optimization problems.