Reference variable methods of solving min-max optimization problems

摘要

In this paper, reference variable methods are proposed for solving nonlinear Minmax optimization problems with unconstraint or constraints for the first time, it uses reference decision vectors to improve the methods in Vincent and Goh (J Optim Theory Appl 75:501-519,1992) such that its algorithm is convergent. In addition, a new method based on KKT conditions of min or max constrained optimization problems is also given for solving the constrained minmax optimization problems, it makes the constrained minmax optimization problems a problem of solving nonlinear equations by a complementarily function. For getting all minmax optimization solutions, the cost function f(x, y) can be constrained as M_1 < f(x, y) < M_2 by using different real numbers M_1 and M_2. To show effectiveness of the proposed methods, some examples are taken to compare with results in the literature, and it is easy to find that the proposed methods can get all minmax optimization solutions of minmax problems with constraints by using different M_1 and M_2, this implies that the proposed methods has superiority over the methods in the literature (that is based on different initial values to get other minmax optimization solutions).