Weak Sharpness For Set-valued Variational Inequalities And Applications To Finite Termination Of Iterative Algorithms

The finite termination of an arbitrary algorithm for solving a classical variational inequality has been studied by many authors. All of them have something in common:the solution set of the classical variational inequality is assumed to be weakly sharp.The set-valued variational inequality as one of the important generalizations of classical variational inequality finds various applications in optimization and control,nonlinear complementary problems. However, there are few results on the finite termination of an arbitrary algorithm for solving a set-valued variational inequality. In this dissertation, we study the weak sharpness for set-valued variational inequalities and established the finite termination of iterative algorithms. These obtained results extend some known results of classical variational inequalities. The main contents of this dissertation are as follows.In the first chapter, we introduce the background and the main work of this dissertation.In the second chapter, we recall some related concepts and conclusions which are useful in main results.In the third chapter, we define a weaker notion of weak sharpness for set-valued variational inequalities in the n-dimensional Euclidean space and present some characterizations of this notion. We also give some examples to illustrate this new notion. Under the assumption of weak sharpness, by using the inner limit of a set sequence we establish a sufficient and necessary condition to guarantee the finite termination of an arbitrary algorithm for solving a set-valued variational inequality involving a maximal monotone map.In the fourth chapter, as applications, we use obtained results to establish the finite termination of the hybrid projection-proximal point algorithm for solving a set-valued variational inequality.