Abstract: In this paper, we present a modified trust-region filter algorithm to solve  the unconstrained minimax problem. The algorithm solves an SQP subproblem to acquire the attempted step. The filter technique is used to weigh the effect of the attempted step so as to avoid the use of a penalty function.  Based on the idea of the latest reference,  we use the Lagrange function as a merit function,  also combine it with the nonmonotone technique to improve the effect of the algorithm. Under some mild conditions, we prove the global convergence and superlinear local convergence. Numerical results show the effectiveness of the algorithm.

Key words: minimax problem, nonmonotone, global convergence, Filter methods, superlinear convergence