Inexact smoothing method for large scale minimax optimization

A new algorithm for the solution of large scale minimax problems of a finite number of functions is introduced. The algorithm is a smoothing method based on a maximum entropy function and an inexact Newton-type algorithm for its solution. Under mild assumptions, only the approximate solution of a linear system is required at each iteration. The algorithm is shown to both globally and superlinearly convergent. Meanwhile some implementation techniques taking advantage of the sparsity of the Hessians of the functions and alleviating the disadvantage effect of the ill-conditioned matrix are considered. Numerical results show that the inexact method is considerably efficient.