Penalty decomposition derivative free method for the minimization of partially separable functions over a convex feasible set

Abstract: In this paper, we consider the problem of minimizing a smooth function, given as finite sum of black-box functions, over a convex set. In order to advantageously exploit the structure of the problem, for instance when the terms of the objective functions are partially separable, noisy, costly or with first-order information partially accessible, we propose a framework where the penalty decomposition approach is combined with a derivative-free line search-based method. Under standard assumptions, we state theoretical results showing that the proposed algorithm is well-defined and globally convergent to stationary points. The results of preliminary numerical experiments, performed on test problems with number of variables up to thousands, show the validity of the proposed method compared with a standard derivative-free line search algorithm. Moreover, it is shown that the method is easily parallelizable and hence capable of taking advantage of parallelization of computation, when possible.