Splitting Algorithms And Parallel Projection Method And Its Application Limited Memory

With the rapid development of science and technology, convex programming and variational inequality problem play very important roles in a lot of problems, e.g., mathematical programming, network economics, transportation research, game theory and regional sciences. How to design efficient algorithms to solve these prob-lems has been a hot research topic for many years.This thesis studies an augmented-Lagrangian-based parallel splitting method to solve the network resource allocation problems and limited-memory projection method for solving variational inequality problems. We transform the large scale problems into multiple subproblems by augmented Lagrangian method. Projection methods are simple and effective methods for solving variational inequality problems. In this thesis we design a new algorithm by improving the search direction.We present an augmented-Lagrangian-based parallel splitting method to solve network resource allocation problems in the chapter2, which are important con-strained optimization problems. The convergence of the method is established under some assumptions. We report some preliminary numerical results, which demon-strate that the method is reliable and efficient in practice.In the chapter3, we present a limited-memory projection method for solving monotone variational inequality problems with simple constrains. At the kth iter-ation, the search direction dk is constructed by combining a profit direction of the problem and the directions used in the last (m-1)th iteration. The method can be viewed as generalization of conjugate gradient methods for solving suitable condi-tions nonlinear programs. The convergence of the method is established under some assumptions, and numerical results demonstrate the method is reliable and efficient in practice.