New Sequential and Parallel Derivative-Free Algorithms for Unconstrained Minimization

This paper presents sequential and parallel derivative-free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest. It is proved that, under mild assumptions, a sufficient decrease condition holds for a nonsmooth function. Based on this property, the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value. If the function is strictly differentiable at its limit points, a (sub)sequence of points generated by the algorithm converges to a first-order stationary point (∇f(x) = 0). If the function is convex around its limit points, convergence (of a subsequence) to a point with nonnegative directional derivatives on a set of search directions is ensured. Preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods.