Study On Some Algorithms For Variational Inequalities And Linearly Constrained Splitting Optimization Problems

In this thesis, we mainly investigate the numerical algorithms, including the projection method for the variational inequalities, and the alternating direction method for the linearly constrained splitting optimization problems. This thesis is divided into seven chapters and organized as follows:In Chapter 1, firstly, we describe the development and current researches on the topic of the projection method for the variational inequalities, and the alternating direction method for the linearly constrained splitting optimization problems. Then, we give the motivation and the main research work of this thesis.In Chapter 2, we introduce some basic notions, definitions, and propositions, which will be used in the sequel. Then we give the evaluation criterions whether the algorithm is good or not.In Chapter 3, we study the projection method for the nonlinear equations. The algorithm is devised based on the the conjugate gradient method. The method can avoid the case that the algorithm may generate the small stepsize consecutively by employing the self-adaptive technique. It inherits the advantages of the conjugate gradient method, the projection method and the self-adaptive method. The numerical results illustrate that the method is effective for the nonlinear equations.In Chapter 4, we investigate the projection method for the variational inequalities. We devise an algorithm for the problems, of which the constrained sets are complex. We use an appropriate hyperplane to separate the current iterate point from the solution set of the problem, which can reduce the difficulty of computing the projection. The procedure of instructing the hyperplane requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. The numerical results illustrate that the method is effective for the variational inequality, especially for the problems with the complex constrained sets.In Chapter 5, we study the alternating direction method for the multiple-sets split feasibility problems. Based on the penalty function, we devise a method for this kind of problems. The algorithm has the global convergence. As the penalty parameter dynamically adjusts, the algorithm can reduce the difficulty of choosing the original penalty parameter. The numerical results illustrate that the method is effective for the multiple-sets split feasibility problems.In Chapter 6, we investigate the alternating direction method for the linearly constrained nonconvex optimization problems. Based on the Kurdyka-?ojasiewicz property of the nonconvex function, we prove the convergence of the method. When the initial point satisfies some mild conditions, we give the local convergence to global minima of the algorithm. The numerical results illustrate that the method is effective for the linearly constrained nonconvex optimization problems.In Chapter 7, the results of this thesis are summarized. And some problems which are thought over in future are put forward.