Some Algorithms For Solving Finite, Semifinite And Generalized Semifinite Minimax Problems

In this paper, quasi-Newton methods and element updating Newton-like algorithms for finite minimax problems, element updating Newton-like algorithms for semifinite minimax problems and a globally convergent method for generalized semifinite minimax problems are studied.At first, we consider the finite minimax optimization problem (MMP)where q = {1,2, ? ? ?, q}, ?_j ∈ q, f^j : R^? → R are Lipschitz continuously differen-tiable.E.Polak et.al. gave a Pironneau-Polak-Pshenichnyi(PPP) algorithm for solving directly finite minimax problems, but the method is only linearly convergent. A quadratically convergent Newton method for finite minimax problems is presented in Ref. 2. To obtain quadratical convergence, however, a strict complementarity at the Danskin point has to be satisfied. The condition is too strict to be satisfied in many practical problems such as semifinite minimax problem being discretized. In Ref. 3, another kind of Newton method for finite minimax problems is presented and, without strict complementarity at the Danskin point, superlinear convergence (of order 3/2) was proven. In this paper, a quasi-Newton method for finite minimax problems is presented, and the globally and locally...