A Smoothing Direct Search Method for Monte Carlo-Based Bound Constrained Composite Nonsmooth Optimization

Abstract

We propose and analyze a smoothing direct search algorithm for finding a minimizer of a nonsmooth nonconvex function over a box constraint set, where the objective function values cannot be computed directly but are approximated by Monte Carlo simulation. In the algorithm, we adjust the stencil size, the sample size, and the smoothing parameter simultaneously so that the stencil size goes to zero faster than the smoothing parameter and the square root of the sample size goes to infinity faster than the reciprocal of the stencil size. We prove that with probability one any accumulation point of the sequence generated by the algorithm is a Clarke stationary point. We report on numerical results from statistics and financial applications.