A Class Of Filter Methods For Solving The Minimax Problem

A lot of project design problems can be transformed into the minimax problem. Some problems in mathematical field,for example,L_∞approximate problem,nonlinear equations,nonlinear constrained optimization,multiobjective programming,etc.,have close relationship with minimax problems.It is a very important research subject in mathematical programming and engineering optimization in recent years.At present,there are three classes of methods for solving minimax problems:direct methods for solving primal problem,smoothing methods for solving it by using some smoothing functions,equivalent methods for solving equivalent constrained optimization. In recent years,Fletcher and Leyffer present filter methods to solve constrained optimization,which numerical tests show that the algorithm is very efficient and reliable.In this paper,a class of filter algorithms is presented for solving the minimax problem,which has the following characters.Firstly,filter technique with trust region frame is used,which avoids using penalty function.Secondly,the subproblem is the standard quadratic programming which is easily solvable.Thirdly,the Hessian matrix of the subproblem is not needed to be positive definitely uniformly,but bounded uniformly.Under the usual assumptions,we analyse the global convergence of the algorithm.Finally,the numerical tests are given,which shows that this new method is very effective.