Trust-region methods for the derivative-free optimization of nonsmooth black-box functions

In this paper we study the minimization of a nonsmooth black-box type function, without assuming any access to derivatives or generalized derivatives and without any knowledge about the analytical origin of the function nonsmoothness. Directional methods have been derived for such problems but to our knowledge no model-based method like a trust-region one has yet been proposed. Our main contribution is thus the derivation of derivative-free trust-region methods for black-box type functions. We propose a trust-region model that is the sum of a max-linear term with a quadratic one so that the function nonsmoothness can be properly captured, but at the same time the curvature of the function in smooth subdomains is not neglected. Our trust-region methods enjoy global convergence properties similar to the ones of the directional methods, provided the vectors randomly generated for the max-linear term are asymptotically dense in the unit sphere. The numerical results reported demonstrate that our approach is both efficient and robust for a large class of nonsmooth unconstrained optimization problems. Our software is made available under request.