An Active Set Smoothing Newton Algorithm For Solving Unconstrained Minimax Problems

In this paper,according to the special structure of the active strategy smoothing max function,the analytic solution is given based on the discussion of polynomial roots.A special case of this active set strategy smoothing equation is the three order equation.When the index set is determined,there is one and only one real root.But when the smoothing parameter becomes more and more smaller,directly using the formula solution will product relative big error.Therefore,the need for solving the formula is modified to improve the calculation accuracy of it.In this paper,we associate the smoothing algorithm and the analytical solution to apply to the unconstrained minimax function.Preliminary experiments show that the number of iterations is less than numerical iteration method.In the smooth Newton method,the smooth function of analytic method and Newton it-eration method is given based on their combination algorithm applied to minimax problem of multicomponent function,which reduce the calculation amount of gradient and Hessian com-ponent functions.Numerical experiments show that the algorithm has some advantages in computational efficiency.