Smoothing and Worst-Case Complexity for Direct-Search Methods in Nonsmooth Optimization

In the context of the derivative-free optimization of a smooth objective function, it has been shown that the worst case complexity of direct-search methods is of the same order as the one of steepest descent for derivative-based optimization, more precisely that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is proportional to the inverse of the threshold squared. Motivated by the lack of such a result in the non-smooth case, we propose, analyze, and test a class of smoothing direct-search methods for the unconstrained optimization of nonsmooth functions. Given a parameterized family of smoothing functions for the non-smooth objective function dependent on a smoothing parameter, this class of methods consists of applying a direct-search algorithm for a fixed value of the smoothing parameter until the step size is relatively small, after which the smoothing parameter is reduced and the process is repeated. One can show that the worst case complexity (or cost) of this procedure is roughly one order of magnitude worse than the one for direct search or steepest descent on smooth functions. The class of smoothing direct-search methods is also showed to enjoy asymptotic global convergence properties. Some preliminary numerical experiments indicates that this approach leads to better values of the objective function, pushing in some cases the optimization further, apparently without an additional cost in the number of function evaluations.