An Aggregate Homotopy Method For Solving Unconstrained Minimax Problems

The minimax problem is a typical nonsmooth optimization problem,and has been widely applied in fields of engineering design,data fitting,structural optimization,optimal control,and so on.Based on the linear homotopy and the aggregate function,an aggregate homotopy method is proposed to solve unconstrained minimax problems.Under some basic assumptions,for almost all points of Rn or a ball set,the existence and the global convergence of a smooth homotopy path are proved.When the homotopy parameter tends to 0,a stationary point of the minimax problem can be obtained.The homotopy parameter in the homotopy mapping is also used as the smoothing parameter of the aggregate function.Therefore,the numerical calculation can take a general path following procedure and does not require a additional adaptive updating strategy of the smoothing parameter.In order to improve the ill-conditioned problem of the aggregate function when the smoothing parameter closes to 0,when the homotopy parameter(the smoothing parameter of the aggregate function)is very small,an alternative strategy of the path following procedure(an endgame strategy)is given,which uses the Newton method to directly calculate the KKT system of the minimax problem.The primarily numerical test results show that the proposed method is efficient and robust.