The Active Set Smoothing Algorithms For Solving Finite-dimensional Unconstrained Minimax Problems

In this paper,a smoothing maximum function with the active set strategy is proposed based on the piecewise quadratic polynomial equation.The piecewise quadratic polyno-mial equation is transformed into a general quadratic polynomial equation by calculating the index set of the component functions related to the smoothing maximum function.Then,by the properties of roots for the quadratic polynomial,a stable calculation strategy for the smoothing maximum function is given.It is proved that the smoothing maximum function is continuously differentiable and its gradient is locally Lipschitz continuous and strongly semi-smooth.The smoothing maximum function is only related to the compo-nent functions whose function values are close to the maximum function value.Hence,it is suitable for the finite-dimensional unconstrained minimax problems with a large num-ber of complex component functions.In order to show the efficiency of the smoothing maximum function,an active set smoothing algorithm based on the smoothing maximum function is proposed for solving the finite-dimensional unconstrained minimax problems with a large number of complex component functions.Preliminary numerical experiments show the feasibility and eficiency of the smoothing maximum function.However,the algorithm above takes a certain amount of time to calculate the function value of the smoothing maximum function and related index set.Therefore,the algorithm cannot guarantee the advantage in computing time to solve the finite-dimensional uncon-strained minimax problems with few component functions or with simple component functions.For this issue,a new active set smoothing function based on the plus function is constructed for the maximum function.It is proved that the smoothing function is continuously differentiable and its gradient is locally Lipschitz continuous and strongly semi-smooth.Then,an active set smoothing algorithm for solving the finite-dimensional unconstrained minimax problems is constructed based on the smoothing function.Prelim-inary numerical experiments show that the algorithm has high computational efficiency to solve the finite-dimensional unconstrained minimax problems.