Study On Extra-gradient Projection Methods For Generalized Variational Inequalities And Quasi-Equilibrium Problem

The generalized variational inequality problem arises in a wide variety of application areas, and it provides very powerful techniques for studying problems arising in mechanics, optimization, transportation, economics equilibrium, contact problems in elasticity, and other branches of mathematics. In this paper,we study several extra-gradient methods for generalized variational inequalities in real Euclidean space and Hilbert space. Moreover, we present an alternative projection method for the quasi-equilibrium problem. The rest of this paper is organized as follows:In Chapter 1, we introduce some basic fundamental knowledge of variational inequalities and generalized variational inequalities, and briefly summarize some known research results. Some basic notions that will be much used are presented in Chapter 2. Properties of projection operators, definitions of monotone multivalued operator, pseudo-monotone multi-valued operator and continuous multivalued operator are introduced in this part.In Chapter 3, we study the generalized variational inequality problem in dimensional real Euclidean space. We extend the extra-gradient projection method proposed for the classical variational inequalities to the generalized variational inequalities. For this algorithm, we first prove its expansion property of the generated sequence with respect to the starting point and then show that the existence of the solution to the problem can be verified through the behavior of the generated sequence. The global convergence of the method is also established under mild conditions.In Chapter 4, we propose an improved two-step extra-gradient algorithm for pseudo-monotone generalized variational inequalities. It requires two projections at each iteration and allows one to take di?erent step-size rules. Moreover, from a geometric point of view, it is shown that the new method has a long step-size,and it guarantees that the distance from the next iterative point to the solution set has a large decrease. Under mild conditions, we show that the method is globally convergent, and then the R-linearly convergent property of the method is proven if a projection-type error bound holds locally.In Chapter 5, a new type of extra-gradient method is presented for generalized variational inequalities with multi-valued mapping in an infinite-dimensional Hilbert space. Related to the infinite sequence generated in the algorithm, the generated sequence possesses an expansion property with respect to the initial point. Then, it is proved that the existence of the solution to the problem can be verified through the behavior of the generated sequence. Under mild assumptions, it is shown that the generated sequence of the method strongly converges to the solution of the problem which is closest to the initial point.In Chapter 6, we extend the current sub-gradient extra-gradient projection method for variational inequalities to generalized variational inequalities. For extra-gradient method, if the feasible set is simple enough, so that projections onto it are easily executed, then this method is particularly useful; but, if the feasible set is a general closed and convex set, this might seriously a?ect the e?ciency of the extra-gradient method. The new sub-gradient extra-gradient method for the classical VI is developed in which replacing the projection region in the second projection by a projection onto a specific half-space, which makes the method more e?cient. Since the GVI is a generalization of the VI,this constitutes the motivation of this chapter. Under suitable conditions, we show that any sequence of the algorithm globally converge to a point of the solution set of the generalized variational inequalities.In Chapter 7, for the quasi-equilibrium problem where the players’ cost and their strategy both depend on the rival’s decisions, we design an alternative extra-gradient projection method for solving it. Di?erent from the classical extra-gradient projection method whose generated sequence has the contraction property with respect to the solution set, the new designed method possesses an expansion property with respect to a given initial point. The global convergence of the method is established under the assumptions of pseudo-monotonicity of the equilibrium function and of the continuity of the underlying multi-valued mapping. Furthermore, we show that the generated sequence converges to the nearest point in the solution set to the initial point. Numerical experiments show the e?ciency of the method.