A Linesearch-based Derivative-free Approach for Nonsmooth Optimization

In this paper, we propose new linesearch-based methods for nonsmooth optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence towards stationary points, using the Clarke-Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the bound-constrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the bound-constrained case to solve the penalized problem, and we prove that every accumulation point of the generated sequence of points is a stationary points of the original constrained problem. In the last part of the paper, we report extended numerical results on both bound-constrained and nonlinearly constrained problems, showing the effectiveness of our approach when compared to some state-of-the-art codes from the literature.