Variable Combination Of Relaxation Algorithms For Variational Inequality Problems

The variational inequality, widely applied in various fields such as Engineering Mechanics, Mathematic Physics, Economic Mathematics, Network Analysis, Cybernetics and Optimization Theory, has become the focus of applied mathematical research in the past decades.Using the idea of combined relaxation method introduced by Konnov, we propose a series of effective and feasible algorithms for solving classical, nonlinear extended, mixed and multi-valued variational inequality problems. These methods usually contain an auxiliary problem that can be used to compute the parameters of a hyperplane which separates the current iterate and solution set. Then in the main iteration process, we project the current iterate onto this hyperplane. We prove that the iterative sequences generated by above methods are Fejér-monotone.In Chapter 2, we propose a suitable method for solving classical variational inequality problem which is proved effective and easy to practice. In addition, we transform a special equilibrium problem to a variational inequality problem. Considering its speciality and combining the former methods, we propose an effective method to solve this problem.In Chapter 3, for nonlinear extended variational inequality, we adjust our auxiliary problem to a nonempty convex and closed-valued mapping, prove the existence of the fixed point of the mapping, which is just the solution of the original problem, and show that the iterative sequence is strongly convergent to a solution of the problem.The mixed and multi-valued variational inequality problems in infinite space are studied in Chapter 4. On the basis of the former algorithms, in view of the speciality of these problems, we propose a combined relaxation method based on a splitting-type technique in the auxiliary problem, which uses a linear search connected with the searching of the Lipschitz constant, (which is different from the step size search used above), and adopts another approach to choose parameters. These modifications make the proving of the convergence of the algorithm more complex, however, we can still prove that the sequences generated by these methods are Fejér-monotone. Besides, we also prove that these sequences are weakly convergent to a solution of the original problem. In addition, we have made a comparison between our methods and other usual ones and finally prove that these methods possess a linear convergence rate under comparatively strong conditions.