Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces

Abstract EN

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces.

Let H1, H2, and H3 be infinite-dimensional real Hilbert spaces, let C⊂H1 and Q⊂H2 be two nonempty closed convex sets, and let A:H1→H3 and B:H2→H3 be two bounded linear operators.

The ASEP in infinite-dimensional Hilbert spaces is to minimize the function fx,y=(1/2)Ax-By22 over x∈C and y∈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 used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such 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).