Study On Stability And Application Of Saddle Point Set Of Real Valued Function

The equilibrium problem provides a unified framework for the study of various variational analysis problems,such as variational inequalities,saddle point problems and minimax problems.Saddle point problem is an important research topic in equilibrium problems,which is widely used in the fields of optimization algorithm,mathematical programming,game theory and so on.On the one hand,this paper mainly studies the stability of the real-valued function ?-approximate saddle point set near the saddle point in the real normed linear space when the feasible region and the objective function are perturbed,including Hausdorff continuity,Holder continuity and Lipschitz continuity.The stability of the real-valued function?-approximate saddle point set is applied to the uncertain two-person zero-sum game.On the other hand,a more general equilibrium problem is studied,and the stability of the approximate solution mapping of the equilibrium problem with parameter scalars in normed linear spaces is obtained,which mainly includes Hausdorff continuity and Lipschitz continuity.In the first chapter,it mainly introduces the development status and research significance of the equilibrium problem and saddle point problem,and then gives the motivation and main work of this paper.In the second chapter,we first describe the mathematical model of the real-valued function perturbed saddle point problem and the definition of the?-approximate solution set,then introduce the related knowledge for the setvalued mapping used in the later proof such as Berge continuity and Hausdorff continuity.And then deriving the Hausdorff continuity of the real-valued function?-approximate saddle point set by using the compactness of the feasible region,the properties of concave-convex functions and Berge continuity.In the third chapter,the definitions of Holder continuity and Hausdorff metric are given at first.Then,the Holder continuity of approximate saddle point set of real-valued functions is obtained by combining the Berge upper semi-continuity of concave-convex functions,Holder continuity and Holder continuity of feasible region.In the fourth chapter,we introduce the mathematical model of uncertain twoperson zero-sum game,and then derive the application of the stability of real-valued function saddle point problem in uncertain two-person zero-sum game according to the conclusions in chapter 2 and chapter 3,and some examples are given to verify its correctness.In Chapter 5,the saddle point problem is extended to the general parametric vector equilibrium problem.In normed linear space,we prove the Lipschitz continuity of approximate solution mapping of the parametric vector equilibrium problem by using the cone concave property of function,Berge upper semi-continuity and the boundedness of approximate solution mapping of parametric vector equilibrium problem.Finally,the full text is summarized,and the future research work is prospected.