The Nonmonotone Filter Methods For Minimax Problems

Optimization plays a important role in our daily life.And many real life can be stated as a minimax problem,such as the problem in engineering optimization design,electronic circuit design,computer aided design,optimal control and countermeasures and other fields.There exist two classes of methods for minimax problems.One is the nonsmooth-type method,including subgradient methods and cutting plane methods.The other is the smoothtype method,including maximum entropy function method,and some methods for non-smooth problems.But in most exist methods,the penalty function is always used as a merit function and the value of objective function is required decreased monotonously.To overcome the above disadvantages,we present two nonmonotone filter methods for minimax problems.One is the modified nonmonotone filter method.It is a method that used the filter set as a criterion to decide whether a trial point is acceptable or not.Therefore,the penalty function is avoided,so does the penalty parameters.Moreover,the modified nonmonotonic technique is adopted in the acceptable criterion.That is to say,the maximum of the value if current objective function and the convex of some pervious values of objective function is calculated first,and then compared with the value of objective function of trial point.If the latter is better than the former,we will accept the trial point as a new iterate.The other method presented in our paper is a nonmonotone flexible filter method for minimax problems.In this method,we constrant a new filter set with a flexible parameter.In update of the parameter is depending on the improvement of the function at the current iteration point.Thus the acceptable criterion of trial point is more relaxed than traditional methods,and Maratos effects can be avoided to a certain degree.Under some reasonable assumptions,the proposed algorithm is globally convergent and the numerical results are given.