AN EFFECTIVE OPTIMIZATION ALGORITHM FOR LOCALLY NONCONVEX LIPSCHITZ FUNCTIONS BASED ON MOLLIFIER SUBGRADIENTS

We present an effective algorithm for minimization of locally nonconvex Lipschitz functions based on mollifier functions approximating the Clarke generalized gradient. To this aim, first we approximate the Clarke generalized gradient by mollifier subgradients. To construct this approximation, we use a set of averaged functions gradients. Then, we show that the convex hull of this set serves as a good approximation for the Clarke generalized gradient. Using this approximation of the Clarke generalized gradient, we establish an algorithm for minimization of locally Lipschitz functions. Based on mollifier subgradient approximation, we propose a dynamic algorithm for finding a direction satisfying the Armijo condition without needing many subgradient evaluations. We prove that the search direction procedure terminates after finitely many iterations and show how to reduce the objective function value in the obtained search direction. We also prove that the first order optimality conditions are satisfied for any accumulation point of the sequence constructed by the algorithm. Finally, we implement our algorithm with MATLAB codes and approximate averaged functions gradients by the Monte-Carlo method. The numerical results show that our algorithm is effectively more efficient and also more robust than the GS algorithm, currently perceived to be a competitive algorithm for minimization of nonconvex Lipschitz functions.