The Projection Algorithms For The Split Feasibility Problem

It is long been optimization theory could always be playing an important role in the development of society and economy, which has been significant for many areas such as infrastructure, economic development, military defense etc. In real life the SFP is a kind of more common optimization problem. As the years past, facing numerous problems in real life, people have been putting forward many optimization algorithm for solving the problems, among which the projection algorithms characterized with simple and straightaway formation is the most basic method.The prime innovations in this article are as follows.(1) Based on projection algorithms in Euclidean space, and wielding the properties of variational inequality problem being equivalent to split feasibility problem, we propose the idea of amended outer gradient algorithms to solving single-set split feasibility problem. Afterwards, we spreaded this algorithm in Hilbert Space. By deducing and proving, the algorithm also has good convergence in Hilbert Space.(2) Based on KM iterative methods for solving the multiple-sets spilt feasibility problem in R space, we spreaded this algorithm in Hilbert Space. And at last, we proved the convergence of the algorithm. By deducing and proving, the multiple-sets spilt feasibility problem has good convergence in Hilbert Space.(3) By using the fact that the variational inequality equals Split feasibility problem in certain conditions, we focus on the more general space-Hilbert space. Later we propose a vital important theory, which could help us study VI and SFP better in Banach space. So we have more choices to solve the relevant problems by taking advantage of the theory.