The Study Of A Class Of Nonlinear Lagrange Method For Constrained Minimax Problems

The constrained minimax optimization problems are common issues in the practical life.A lot of experts and scholars have researched these problems in many aspects,and presented many methods to solve these problems.This thesis proposes effective methods to solve the constrained minimax deterministic and stochastic optimization problem better,based on the previous work.The main works in this thesis are as follows:(1)The thesis constructs a novel nonlinear Lagrange function based on the study of nonlinear Lagrange method for solving the constrained minimax deterministic problems,and establishes a minimization problem which takes the novel nonlinear Lagrange function as the objective function,then proposes a nonlinear Lagrange algorithm.Under some assumptions,this thesis establishes the convergence theory corresponding to the nonlinear Lagrange algorithm which indicates that the optimal sequence solutions obtained by the algorithm is locally convergent to K-K-T of the original problem at a linear rate when the control parameter is less than a threshold.Finally,the nonlinear Lagrange algorithm is programed in Matlab languages based on the 10 typical numerical examples.Numerical experiments for 10 problems are implemented with this algorithm.The results of the numerical experiments prove the feasibility and effectivity of the algorithm.(2)Combined with the basic idea of the sample average approximation method,this thesis constructs a sample average nonlinear Lagrange function based on the above nonlinear Lagrange function for the constrained minimax stochastic optimization problem,and proposes the sample average nonlinear Lagrange algorithm.Under some assumptions,this thesis proves that the optimal sequence solution obtained by the sample average nonlinear Lagrange algorithm is convergent with probability 1 to the optimal solution of the original problem according to the convergence theory of the constrained minimax deterministic problem.Finally,the sample average nonlinear Lagrange algorithm is programed in Matlab languages based on the 8 numerical examples.Numerical experiments for 8 problems are implemented with this algorithm.The results of the numerical experiments preliminarily prove that the sample average nonlinear Lagrange algorithm is feasible and effective.