A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

In this paper, a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints. This new objective penalty function combines the objective penalty and constraint penalty. By the new objective penalty function, a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint. One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence. Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established. Based on these results, an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions. Furthermore, an algorithm is developed for finding a local solution of the constrained minimax problems, with its convergence proved under certain conditions. Preliminary results This research was supported by Natural Science Foundation of Chongqing (Nos. cstc2013jjB00001 and cstc2011jjA00010), by Chongqing Municipal Education Commission (No.KJ120616). J. Li SITE, Federation University Australia, Ballarat, VIC 3353, Australia e-mail: [email protected] J. Li Z. Wu (&) Department of Mathematics, Chongqing Normal University, Chongqing 400047, China e-mail: [email protected] Q. Long School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China 123 J. Oper. Res. Soc. China (2014) 2:93–108 DOI 10.1007/s40305-014-0041-3