An Adaptive Smoothing Method for Continuous Minimax Problems

A simple and implementable two-loop smoothing method for semi-infinite minimax problem is given with the discretization parameter and the smoothing parameter being updated adaptively. We prove the global convergence of the algorithm when the steepest descent method or a BFGS type quasi-Newton method is applied to the smooth subproblems. The strategy for updating the smoothing parameter can not only guarantee the convergence of the algorithm but also considerably reduce the ill-conditioning caused by increasing the value of the smoothing parameter. Numerical tests show that the algorithm is robust and effective.

A Bimaximum Entropy Method for Solving Nonlinear Constrained Continuous Minimax Problem

A numerical method is proposed for solving a sort of constrained continuous minimax problem, in which both the objective function and the constraint functions are continuously differentiable about superior decision variables and are continuous about lower decision variables .Besides,the constraint functions include only superior or lower decision variables．The problem is transformed into unconstrained differentiable problem with the idea of the discrete maximum entropy function and the continuous maximum entropy function and the penalty function method．The basic algorithm is established．The convergence is proofed．Numerical examples are given and show the efficiency and the reliability of the algorithm.