Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces. (7th May 2013) - Chen, Rudong; Li, Junlei; Ren, Yijie
Abstract : We studied 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 ⊂ H 1 and Q ⊂ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3 and B : H 2 → H 3 be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function f x, y = (1 / 2) A x - B y 2 2 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).