Study Of The Projection Algorithm For The Multiple-sets Split Feasibility Problem

The split feasibility problem is an important type of optimization problems, which generatesfrom engineering practice and has important applications in biology, physics, signal processing andimage reconstruction. The multiple-sets split feasibility problem requires looking for a point nearestto a family of non-empty closed convex sets in one space such that its image in a lineartransformation will be nearest to another family of non-empty closed convex sets in the image space.People have proposed many algorithms to solve the multiple-sets split feasibility problem, in whichthe projection algorithm is a kind of basic and important method, because the projection algorithmhas simple structure and can be applied easily.This paper focuses on solving the multiple-sets split feasibility problem with the projectionmethod. The main innovation works are:（1）Based on the projection-type methods for solving the multiple-sets split feasibilityproblem, we propose a new projection algorithm which does not need to calculate the spectralradius of the matrix. The projection algorithm can improve operation efficiency and reducecalculation workload since choosing the stepsize does not need to start from the initial value duringthe iterative process, and it is stable. Global convergence of the enhanced algorithm is shown. Inaddition, we report the computational experience.（2）Based on the inexact projection-type methods for solving the split feasibility problem, wepresent the inexact projection methods to solving the multiple-sets split feasibility problem. Firstly,each iteration of the first proposed algorithm consists of a projection onto a halfspace whichincludes the given non-empty closed convex set. The new algorithm is easy to implement. Secondly,we present modification of the inexact projection method with constant stepsize by adoptingArmijo-like search which does not require the computation of the matrix norm and the largesteigenvalue, and we make another correction by using iterative step as a predict step. At last, weobtain the gobal convergence of this algorithm and give the numerical experiments which show thatthe improved algorithm has good behavior.（3）We furthermore present a self-adaptive inexact projection method by the Krasnoselski-Mann(KM) iteration form, in which the objective function can decrease sufficiently at eachiteration. We proved the convergence of the self-adaptive inexact projection method and performednumerical experiment. The results of experiment show that the new method can be applied easily and has a faster convergence speed.