Trust Region Secant/Finite Difference Method for Large Sparse Minimax Problems

Abstract

A secant/finite difference algorithm based on trust region strategy is presented. This algorithm is designed to solve the minimax optimization problem of a finite number of functions, whose Hessian matrices are normally sparse. By integrating a secant method and a finite difference method, and adopting a symmetrically consistent partition of the columns of the Hessian matrices, the algorithm can employ the gradient evaluations as efficiently as possible to build quadratic approximations to the functions. This technique will, at every iterative step, have m, the number of functions, less the number of gradient evaluations than that of the direct method. And in order to enlarge the region of convergence, the trust region strategy is also adopted. The algorithm is proved to have good global and local convergence properties with q-superlinear convergence and r-convergence rate. The robustness and efficiency of the algorithm is verified by numerical tests, and its performance is comparable to or better than that of other algorithms currently available.