The Study Of A Class Of Nonlinear Lagranee Method For Unconstrained Minimax Problems

This paper studies the nonlinear Lagrange method for solving unconstrained minimax problems. The main work are as follows:(1) For solving a class of unconstrained minimax deterministic problems, we construct a class of nonlinear Lagrangian function and the corresponding algorithmic framework, and a set of mild conditions are proposed to guarantee the convergence theory of the algorithm. The unified convergence analysis framework for the class of algorithm is established. It is shown that the sequence solutions obtained by the class of algorithms are Q-linearly convergent when the controlling parameter is less than a threshold under some assumptions and the error bounds of the sequence solutions are obtained at the same time. Furthermore, it is presented that the proposed class of nonlinear Lagrangians contains four well-known nonlinear Lagrangians for unconstrained minimax problems appearing in the literatures. At last, we programs in Matlab languages based on the four typical nonlinear Lagrangians, and numerical results for ten typical test problems are reported, which show that the four Lagrangians have their own advantages respectively.(2) For solving a class of unconstrained minimax stochastic problems, we construct a sample average approximation minimization problem to approximate the original minimization problem with sample average approximation method, based on a nonlinear Lagrangian function. The" corresponding sample average approximation nonlinear Lagrange algorithm and a set of mild conditions to guarantee the convergence are constructed, and the convergence analysis with probability1for the optimal value and the optimal solution of the SAA problem are presented. Furthermore, combined with the Second-Order Optimality Conditions of deterministic unconstrained minimax problems, the convergence analysis with probability1for the SAA nonlinear Lagrange algorithm is presented. At last, numerical experiments for five concrete problems are implemented with this algorithm, which indicate that the algorithm is promising.