A Sub - Gradient Gradient Algorithm For Quasi - Variational Inequalities

Quasi-variational inequality problem(QVIP) is one of the important research problems in the field of optimization. It has been widely used in economy, engineering, optimization and automatic control and other fields. Therefore, studying the effective numerical method of quasi-variational inequality has important theoretical significance and practical value. It has been widely concerned by many domestic and foreign experts and scholars since being proposed. However, the study of the QVI to date is infance at best. So, to find and design algorithms for solving quasi-variational inequality problem is a more meaningful study. Among them, the projection-type algorithms is the most representative. The algorithm has following distinct advantages.:if the constraint is easily calculated, the algorithm is easy to perform; besides, the store capacity of algorithm is small and it can be used for solving large-scale problems. As we all know, in some cases, it is not a simple problem to calculate the projection from a point to a feasible set. Sometimes, it is impossible or needs too much work to exactly compute the orthogonal projection. And when these conditions occur, the projection-type method may be affected. While, relaxed projection algorithm can overcome this problem, which greatly reduce the amount of calculation and the difficulty of the traditional projection algorithm. However, the difficulty of relaxed projection algorithm is that, the projected areas would change along with the iteration points, because the structure of the projection area need information about current or previous iteration point. At present, the relaxed projection algorithm has been attracted by many domestic and foreign experts and scholars, which has obtained certain results. But, it also has its shortcomings that we need struct the hyperplane at each iterate, and then calculate the subgradient of some function. As we know it is not an easy problem, which restricts the feasibility and validity of this algorithm. In this paper, we design subgradient extragradient projection algorithms to solve the quasi-variational inequality problem. At correction step, we replace the closed convex set with a half space. Futhermore, we avoid to calculate the subgradient successfully, which in some extent, reduces the difficulty of calculation.The article structure arrangement is as follows:The first chapter is an introduction. we mainly describe the specific definition, application backgrounds and the research situations of the quasi-variational inequality problems(QVIP) and the main results obtained in the article.In the second chapter, we proposed the subgradient extragradient projection algorithm of the quasi-variational inequality problem. In order to overcome the difficulty on the computation of the orthogonal projection, we proposed a kind of subgradient extragradient projection algorithm with fixed step length, which could avoid calculate the subgradient. And for the study of the quasi-variational inequality, it’s not yet a kind of computational less difficulty algorithm. At the end of this chapter, we give an example to illustrate the feasibility and validity of algorithm.The third chapter is the improvement of the algorithm of chapter 2. We proposed a changed step length subgradient extragradient projection algorithm. It is known to all, when we prove the problem’s convergence, we need to assume that the map F is Lipschtiz continuous. In order to overcome the strong condition, we improved the first algorithm and replace the fixed step with Amijio-like variational stepsize in the original algorithm, which enlarge the use range of algorithms. In addition, we do some numerical examples to illustrate the practicability and validity of algorithm.