Linear Convergence Analysis Of The Split Equation Problem

In 2012,A.Moudafi proposed the split equality problem SEP.In order to solve SEP,A.Moudafi introduced the alternating CQ-algorithm and the relaxed alternating CQ-algorithm.Notice that these algorithms are all weak convergence in real Hilbert spaces.In 2012,A.Moudafi proposed the split equality null point problem SENP,and also introduced the algorithm to solve SENP.After that,Chen et al.,H.Zegeye and M.Eslamian proposed and studied the strong convergence of the general split equality problem GSEP and the split equality common null point problem SECNP in real Hilbert spaces.Recently,Shi et al.obtained the linear convergence of SEP by using gradient projection algorithm.The main purpose of this paper is to study the linear convergence of the general split equality problem and the split equality common null point problem.In order to analyze the linear convergence of these two kinds of problems,we use the concept of bounded linear regularity.Firstly,in order to solve the general split equality problem,we propose a gradient projection algorithm and show the linear convergence of the sequence generated by the algorithm to a solution of GSEP.Second,for the split equality common null point problem,we study an iterative algorithm and prove that the sequence generated by the algorithm converges linearly to a solution of SECNP.As a special case,in this part we will use the result to study the split equality optimization problem SEOP.In each part,we give numerical examples related to these two kinds of problems.