An adaptive nonmonotone trust-region method with curvilinear search for minimax problem

In this paper we propose an adaptive nonmonotone algorithm for minimax problem. Unlike traditional nonmonotone method, the nonmonotone technique applied to our method is based on the nonmonotone technique proposed by Zhang and Hager [H.C. Zhang, W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14(4)(2004) 1043–1056] instead of that presented by Grippo et al. [L. Grippo, F. Lampariello, S. Lucidi, A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal. 23(4)(1986) 707–716]. Meanwhile, by using adaptive technique, it can adaptively perform the nonmonotone trust-region step or nonmonotone curvilinear search step when the solution of subproblems is unacceptable. Global and superlinear convergences of the method are obtained under suitable conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm.