Projection-typealgorithmfor Variational Inequality With Multi-valued Mapping

In this dissertation, some topics, such as a new projection algo-rithm for variational inequality with multi-valued mapping, a double projectionalgorithm for variational inequality with multi-valued mapping, modified ex-tragradient method for variational inequality with multi-valued mapping, meritfunctions for variational inequality with multi-valued mapping, are investigated.This dissertation is divided into four chapters.In chapter 1, we introduce a projection algorithm for variational inequal-ity with multi-valued mapping. First, we prove that the hyperplane separatesstrictly current iterate and the solution set. Moreover, our method is provento be globally convergent to a solution of the variational inequality problem,provided the multi-valued mapping is continuous and pseudomonotone withnonempty compact convex values. Finally, we present some numerical experi-ments for the proposed algorithm.In chapter 2, we propose a double projection algorithm for variationalinequality with multi-valued mapping. Our method is proven to be globallyconvergent to a solution of the variational inequality problem, provided themulti-valued mapping is continuous and pseudomonotone with nonempty com-pact convex values. We provide a result on the convergence rate of the iterativesequence generated by the algorithm under the assumption of a certain errorbound to hold locally. A class of algorithmic framework for solving the varia-tional inequality is also established. Finally, numerical results are reported.In chapter 3, we propose a modified extragradient method for variationalinequality with multi-valued mapping. We prove the convergence of the itera-tive sequence generated by the algorithm if the solution set of the variational inequality is nonempty. We also prove that the solution set of the variationalinequality is empty if and only if the sequence generated by the algorithm di-verges. Finally, preliminary computational experience is reported.In chapter 4, we consider three classes of merit functions of variationalinequality with multi-valued mapping. We prove that the zeros of these meritfunctions are precisely solutions of the variational inequality. Using thesefunctions, we obtain error bounds for the solution of variational inequalityunder some mild conditions.