Primal Dual Type Algorithms For Solving A Class Of Nonlinear Minimax Problems

Primal dual algorithm is always of prime importance in the process of solving minimax problems.With a broad spectrum of applications of minimax problems in various fields,such as optimal control,operations management,transportation and engineering design,primal dual algorithm captures widespread concerns from researchers.A class of related algorithms have been proposed with analysis and applications.Primal dual algorithm updates primal and dual variables respectively at each iteration,and the coupled mode between primal variable and dual variable is linear.However,the researches of related primal dual methods for nonlinear coupled mode between primal and dual variables are scarce and ambiguous.This thesis present related primal dual type algorithms for a class of nonlinear minimax problems.Firstly,nonlinear primal dual algorithm(I)was proposed for solving a nonconvexconcave nonlinear minimax problem.It updates the dual variable by the linearization and extrapolates it with stepsize 1,then updates the primal variable.With the help of monotone operator theory,the optimality condition of problem was described as the inclusion problem of maximal monotone operator for theoretical analysis.The local convergence of nonlinear primal dual algorithm(I)was established in contraction framework under metric regularity and other mild conditions.And convergence rate is O(1?~2).The results of numerical experiments show that nonlinear primal dual algorithm(I)is efficient with high accuracy for solving the inverse correlation nonlinear problem in distributed parameter identification of differential equations.Secondly,nonlinear primal-dual algorithm(?)was raised for solving a nonconvexstrongly concave nonlinear minimax problem.Its iteration order is different from nonlinear primal-dual algorithm(I),which updates the primal variable by the linearization firstly,and extrapolates it with a proper stepsize,then updates the dual variable normally.The convergence was analyzed under strong convexity of objective function and Lipschitz property of coupling term.The local convergence of nonlinear primal dual algorithm(?)was established with some conditions for modulus and convergence rate is O(1/?~2).