Weak-sharp Condition And Iterative Algorithms For Mixed Variational Inequalities

In this dissertation,we present a variable metric inertial proximal point algorithm for solving general mixed variational inequality problem,and an equivalent characterization of the solution set of the generalized mixed variational inequality problem satisfying the weak-sharp condition,and finite convergence of any iterative algorithm under the condition that the solution set satisfy the weak-sharp.The thesis is divided into three chapters,the specific content is as follows:In chapter 1,we introduce the research status,present situation and the main contents of this thesis.In chapter 2,firstly,we present a variable metric inertial proximal point algorithm for solving general mixed variational inequality problem.Our algorithm consists of classic proximal point method and inertial method.At each step of the algorithm,we use the variable positive definite metric which could define a variable metric.Secondly,under suitable assumptions,we establish the global convergence and nonasymptotic O(1/k)convergence rate result for the proposed algorithm.Finally,we show that the relation between the empty solution set of the general mixed variational inequality and the unbounded sequence obtained by our algorithm.In chapter 3,firstly,we propose a new weak-sharp condition for the solu-tion set of the generalized mixed variational inequality problem.We obtain the equivalent characterizations of weak-sharp condition by some properties of the support function of the constraint set.In addition,under the condition that the solution set of the generalized mixed variational inequalities satisfied the weak-sharp,we obtain equivalent conditions of the arbitrary finite convergent iterative algorithm.Finally,we take an special example of the hybrid projection-proximal point algorithm for the generalized mixed variational inequalities,obtaining the finite convergence of the algorithm under certain conditions.