Study On Proximal Point Algorithms For Nonconvex-Nonconcave Saddle Point Problems

Saddle point problems are very important in the research of mathematical programming and game theory.They provide an effective expression and basic tool for the research of minimax problems,Lagrange dual problems,variational inequality,Nash equilibrium problems,and in optimization algorithm,game theory,mathematical programming,Machine learning and other fields are widely used.On the one hand,this thesis studies the sufficient conditions for the existence of the solution of saddle point problems and the solution set to be a closed set in the case of quasiconvex-quasiconcave of the objective function.On the other hand,a new proximal point algorithm is proposed to solve nonconvex-nonconcave saddle point problems with faster convergence speed and smaller error.The main achievements of this article are as follows:1.We mainly introduce the research significance and development of saddle point problems and their optimization algorithms,as well as the concrete achievements of scholars at home and abroad on the existence of solution set of saddle point problems and related optimization algorithms in recent years,and describe the motivation and main research work of this thesis.2.The concepts of asymptotic function and the uniformly-same order function are introduced.The existence of the solution of saddle point problems when the solution set is a compact convex set obtained by Karamadian through the quasiconvex and quasi-convex assumption of the objective function is extended to the existence of the solution when the solution set is a closed convex set.3.We study the iteration of the classical proximal point algorithm.Under the condition of considering the average iteration points,we have obtained the relevant convergence conclusion through appropriate assumptions.4.A new proximal point algorithm is proposed to solve strongly quasiconcexstrongly quasiconcave saddle point problems,and under appropriate assumptions,the convergence analysis is carried out to ensure that the sequence generated by the proximal point algorithm for the nonconvex-nonconcave saddle point problems can converge to the unique Nash equilibrium point,and the effect of the algorithm is demonstrated through numerical experiments.Finally,we summarize the results of this thesis and make some discussions.