A heuristic iterated-subspace minimization method with pattern search for unconstrained optimization

Recently, an increasing attention was paid on different procedures for an unconstrained optimization problem when the information of the first derivatives is unavailable or unreliable. In this paper, we consider a heuristic iterated-subspace minimization method with pattern search for solving such unconstrained optimization problems. The proposed method is designed to reduce the total number of function evaluations for the implementation of high-dimensional problems. Meanwhile, it keeps the advantages of general pattern search algorithm, i.e., the information of the derivatives is not needed. At each major iteration of such a method, a low-dimensional manifold, the iterated subspace, is constructed. And an approximate minimizer of the objective function in this manifold is then determined by a pattern search method. Numerical results on some classic test examples are given to show the efficiency of the proposed method in comparison with a conventional pattern search method and a derivative-free method.