An Active Set Smoothing Method for Solving Unconstrained Minimax Problems

In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Under the boundedness assumption on the level set of the objective function, the convergence of the proposed method is established. Numerical experiments show that the proposed method is feasible and efficient, particularly for the minimax problems with very many component functions.