On Subgradient Extragradient Projection Algorithms For Solving Variational Inequality Problems

Variational inequality problem(VIP) is a very important research field in the Operational Research and has a wide range of applications in signal processing, image reconstruction, system identification, automatic control and other science fields. It has been concerned widely by many domestic and foreign experts and scholars since being proposed. Some fruitful results have been achieved in its theories and algorithms. Among others, the projection-tye algorithms are impor-tant and representative in these algorithms. It is well known, if the projection is easily calculated, projection-type algorithms are not only simple in form, but also practical and effective, small space occupied, suitable for dealing with large scale problems. However, in some cases it is impossible or needs too much work to exactly compute the orthogonal projection, which will certainly affect the conver-gence speed. To overcome this shortcoming of the projection algorithm, Censor presented the subgradient extragradient projection algorithm. To some extent, he has reduced the computational complexity of the algorithm. On this basis, we give a further study for the subgradient extragradient projection algorithm in this article. The full thesis is divided into three chapters.The first chapter is an introduction. We describe the application back-grounds and the research situations of the variational inequality problems (VIP) and the main results obtained in this article.In the second chapter, we propose two Armijo-like stepsize subgradient ex-tragradient projection algorithms and their inexact forms for solving variational inequality problems. In order to overcome the difficulty on the computation of the orthogonal projection, Censor proposes a class of subgradient extragradient projection algorithms, which replace the closed convex set with a special structure half-space. In the convergence analysis, he needs to assume that the map F is Lipschitz continuous. To remove this strong condition, we improve the subgradi-ent extragradient projection algorithms that proposed by Censor. We replace the fixed step with Armijo-like variable stepsize in the original algorithm, and show the convergence without the assumption that the map F satisfies Lipschitz conti-nuity. The condition of convergence in these algorithms is weakened. Therefore, we enlarge the use range of algorithms. In addition, we also propose the inexact forms of the corresponding algorithms. Compared with the original algorithms, the inexact algorithms can produce more iteration points, which make us choose the iteration points more flexible. Therefore, the proposed algorithms in this chapter have some significance and value.In the third chapter, we give the two-subgradient extragradient projection algorithm convergence analysis that proposed by Censor. Taking the half-space structure in the second chapter is too special into account, Censor proposed the two-subgradient extragradient projection algorithm. In this algorithm, two pro-jections are cast to the half-space. Unfortunately, he only proved the bounded-ness of the iterative sequence{xk} and did not give the proof of the convergence of the algorithm. In this chapter, we show the global convergence of the two-subgradient extragradient projection algorithm, which makes up the deficiency of the algorithm to some extent.