Regularization method for the approximate split equality problem in infinite-dimensional Hilbert spaces (Q370230)

Summary: We study the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let \(H_1\), \(H_2\), and \(H_3\) be infinite-dimensional real Hilbert spaces, let \(C \subset H_1\) and \(Q \subset H_2\) be two nonempty closed convex sets, and let \(A : H_1 \to H_3\) and \(B : H_2 \to H_3\) be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function \(f(x, y) = (1/2)||Ax - By||^2_2\) over \(x \in C\) and \(y \in Q\). Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we use the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and show that the sequence generated by such an algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking \(B = I\) in the ASEP, we recover the approximate split feasibility problem (ASFP).