Algorithms For Set-Valued Variational Inclusions And Their Applications

Variationai inequality theory has emerged as an important branch of applied mathematics with a wide range of applications in industry, physical, regional, social, pure and applied sciences. In recent years, the research on the algorithms for all kinds of variationai inequalities (inclusions) have been paid so much attention that it is necessary to do further research on the theory, construction and application of the algorithms, the research on which is closely related to such mathematical branches as linear and nonlinear analysis, geometry in Banach spaces and numerical computation. Therefore, the research for them has certain degree of difficulty. In this paper, we discuss the existence of the solution, iterative algorithm, and error bounds for a class of variationai inequalities (inclusions) in R"paces, Hilbert spaces, Banach spaces, and metric spaces, respectively. Furthermore, the applications of set-valued variationai inclusions in optimal control and dynamical systems are also discussed. Details are as follows:1. A development survey for variationai inequalities is presented.2. A modified prediction-correction method for solving (general) monotone linear and nonlinear variationai inequality problems is proposed. A practical and robust prediction stepsize choice strategy is developed, which needs only a projection for each line-search procedure. The global convergence of the proposed algorithm is established under the same conditions used in the original prediction-correction method. Numerical results and comparison with some projection-type methods for improving Korpelevich extragradient method are also given to illustrate the efficiency of the proposed method.3. The equivalence between generalized set-valued mixed variationai inequalities in Hilbert spaces and fixed point problems is established by using a new and innovative technique. Based on the equivalence, a class of iterative algorithms for solving generalized set-valued mixed variationai inequalities and related optimization problems are suggested and analyzed. A class of generalized general mixed quasi variationai inequalities in Hilbert spaces are introduced and studied. The existence of the solution of the auxiliary problem for the generalized general mixed quasi variationai inequalities is proved, a predictor-corrector method for solving the generalized general mixed quasi variationai inequalities is suggested and analyzed by using the auxiliary principle technique. A new proximal point algorithm for set-valued variationai inclusions in Hilbert spaces is suggested, and its convergence needs only the monotone operator. Theconcepts of the global (local) resolvent-type error bounds for set-valued quasi variational inclusions in Hilbert spaces are presented, which can be used to analyze the convergence rates of various methods. The global resolvent-type error bounds for generalized quasi-variational inclusions are given under certain conditions.4. A new algorithm for the set-valued quasi-variational inequalities in a uniformly smooth Banach spaces is analyzed, which corrects and improves Noor's results. A class of new set-valued variational inclusions in Banach spaces are introduced and studied, which include many variational inclusions studied by others in recent years. By using some new and innovative techniques, several existence theorems for the generalized set-valued variational inclusions are given, and some perturbed (one-step and three-step) iterative algorithms are suggested under the setting of the smooth and uniformly smooth and q-uniformly smooth Banach spaces. The concept of midpoint locally K-uniformly smooth space is introduced, and its properties and relations between it and other known various K-smooth spaces are discussed.5. Several inequalities of Frechet spaces are given in this paper, which can be regarded as the Frechet spaces versions of the well-known polarization identity occurring in Hilbert spaces. The inequalities developed here have various applications in a number of fields. By using these inequalities, many known resu...