An Active Set Smoothing Maximum Function And Its Application

In this paper,for the finite-dimensional unconstrained minimax problem with a large number of complex component functions,an active set smoothing maximum function is proposed based on piecewise cubic polynomial equation,given the direct calculation method for the index set,and based on the index set,the smoothing equation is trans-formed into a general cubic polynomial equation,the index set is determined,the equation has only one real root and its single,based on the properties of the roots of the cubic polynomial equation,give a stable calculation strategy of an active set smoothing max-imum function.Set of the direct calculation method of index set,the stable calculation strategy of an active set smoothing maximum function,combined with the armijo line search,the negative gradient direction and newton direction,and the update strategy for smoothing parameter,its application to solving the finite-dimensional unconstrained minimax problems with large component functions,preliminary numerical experiments show the efficiency of the proposed method.For the general finite-dimensional unconstrained minimax problem,a twice continu-ously differentiable and active set smoothing maximum function is constructed based on the cubic plus function,combine armijo line search strategy,newton direction,negative gradient direction and update strategies for smoothing parameter and scaled parameter,its application to solving the finite-dimensional unconstrained minimax problems,where the smoothing parameter update strategy uses geometric update,the scaling parameter are selected from the bounded regions.Since the gradient and hessian matrix of the active set smoothing maximum function are only related to the gradient and hessian matrix of the partial component functions,therefore,during the iteration of the algorithm,the num-ber of gradient and hessian matrix calculations is dramatically reduced,and hence the computation cost is greatly reduced especially for the finite-dimensional unconstrained minimax problems with large number of component functions and complicated gradi-ents or hessians matrix,the algorithm is more efficient,and numerical results show the efficiency of the proposed algorithm.