Merit Functions And Error Bounds For Constrained Mixed Set-valued Variational Inequalities Via Generalized F-Projection Operators In Hilbert Spaces

The variational inequality is an important branch of mathematics. With the rapid development of scientific research and production development and electronic computer, constrained variational inequality problems have found many applications in many kinds of problems. In this dissertation, we respectively introduce and investigate a constrained mixed set-valued variational inequality(for short, MSVI) and a constrained mixed inversed variational inequality(for short, MIVI) in Hilbert spaces. This dissertation will maily study the merit functions of MSVI and MIVI by the generalized f- projection operator. The main contents are as follows:In the first chapter, the related background of the variational inequality and the main work of this dissertation are mainly summarized. We introduce a constrained mixed set-valued variational inequality(for short, MSVI) and a constrained mixed inversed variational inequality(for short, MIVI).In the second chapter, we first give some definitions and some lemmas.We also show that MIVI is a special case of MSVI. In addition, we introduce the generalized f-projection operator and some properities, which is the main tool used in this dissertation, and it is also an extension of the approximation operator. Finally, the solution set of MSVI and MIVI was respectively shown to be a singleton under strict monotonicity.In the third chapter, we propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e., upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity.In the fourth chapter, by using the results obtained in the third chapter, we propose four merit functions for the constrained MIVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e., upper estimates for the distance to solutions of the constrained MIVI under strong monotonicity and Lipschitz continuity.In the fifth chapter, we summarize the contents of this dissertation and give some problems.