An optimization approach to solving the split feasibility problem in Hilbert spaces
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An optimization approach to solving the split feasibility problem in Hilbert spaces Simeon Reich1 · Truong Minh Tuyen2 · Mai Thi Ngoc Ha3 Received: 11 April 2020 / Accepted: 26 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem we introduce two iterative methods by using an optimization approach. Our iterative methods do not depend on the norm of the transfer operators. Keywords Hilbert space · Metric projection · Nonexpansive mapping · Split feasibility problem Mathematics Subject Classification 47H05 · 47H09 · 49J53 · 90C25
1 Introduction Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces H1 and H2 , respectively. Let T : H1 −→ H2 be a bounded linear operator and let T ∗ : H2 −→ H1 be its adjoint. The split convex feasibility problem (SCFP, for short) is formulated as follows: Find an element x ∗ ∈ C such that T x ∗ ∈ Q.
(1.1)
The SCFP was first introduced by Censor and Elfving [1] for modeling certain inverse problems. It plays an important role in medical image reconstruction and in signal processing (see [2,3]). Since then, several iterative algorithms for solving (1.1) have been proposed and analyzed. See, for example, [2–16] and references therein.
B
Truong Minh Tuyen [email protected] Simeon Reich [email protected] Mai Thi Ngoc Ha [email protected]
1
Department of Mathematics, The Technion – Israel Institute of Technology, 32000 Haifa, Israel
2
Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Thái Nguyên, Vietnam
3
Thai Nguyen University of Agriculture and Forestry, Thái Nguyên, Vietnam
123
Journal of Global Optimization
Reich et al. [17] have recently presented and studied the following split feasibility problem with multiple output sets in Hilbert spaces: Let H , Hi , i = 1, 2, . . . , m, be real Hilbert spaces and let Ti : H −→ Hi , i = 1, 2, . . . , m, be bounded linear operators. Let C and Q i be nonempty, closed and convex subsets of H and Hi , i = 1, 2, . . . , m, respectively. Suppose m T −1 (Q )) = ∅. We consider the following problem: that S = C ∩ (∩i=1 i i Find an element x † ∈ S,
(1.2)
that is, x † ∈ C and Ti x † ∈ Q i for all i = 1, 2, . . . , m. More precisely, this problem calls for devising an iterative algorithm for constructing an element in S. In order to solve Problem (1.2), Reich et al. [17] introduced the following two iterative methods: For any x0 , y0 ∈ C, let {xn } and {yn } be two sequences generated by m ∗ xn+1 = PC xn − γn Ti (I − PQ i )Ti xn , (1.3) i=1
yn+1 = αn f (yn ) + (1 − αn )PC yn − γn
m
Ti∗ (I
− PQ i )Ti yn ,
(1.4)
i=1
where f : C −→ C is a strict contraction mapping H1 into itself with the contraction coefficient c ∈ [0, 1), {γn } ⊂ (0, ∞) and {αn } ⊂ (0, 1). They established the weak and strong convergence of the sequences generated by iterative methods (1.3) and (1.4), respectively. To this end, they had to to u
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