Solving the multiple-set split feasibility problem and the equilibrium problem by a new relaxed CQ algorithm
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Solving the multiple-set split feasibility problem and the equilibrium problem by a new relaxed CQ algorithm Kunrada Kankam1 · Pittaya Srinak1 · Prasit Cholamjiak1 · Nattawut Pholasa1 Received: 8 May 2018 / Accepted: 28 September 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we introduce a new kind of a relaxed CQ algorithm to find the solution of the multiple-set split feasibility problem and the equilibrium problem in a Hilbert space. We prove weak and strong convergence theorems to the proposed algorithm under some mild conditions. Finally, we provide some numerical experiments to show the efficiency and the implementation of our method. Keywords Multiple-set split feasibility problem · Equilibrium problem · Relaxed CQ algorithm · Hilbert space · Weak and strong convergence · Bounded linear operator Mathematics Subject Classification 65K05 · 90C25 · 47J25
1 Introduction Let H and K be real Hilbert spaces. Let C and Q be nonempty, closed and convex subsets of H and K , respectively. In this work, we study the multiple-set split feasibility problem (MSFP) in Hilbert spaces. This problem is to find a point x ∗ such that x ∗ ∈ C :=
t i=1
Ci ,
Ax ∗ ∈ Q :=
r
Q j,
(1.1)
j=1
t and {Q j }rj=1 are closed convex subsets where t ≥ 1 and r ≥ 1 are given integers, {Ci }i=1 of R N and R M , respectively and A is a given M × N real matrix (A∗ is the transpose of A).
B
Nattawut Pholasa [email protected] Kunrada Kankam [email protected] Pittaya Srinak [email protected] Prasit Cholamjiak [email protected]
1
School of Science, University of Phayao, Mae Ka, Phayao 56000, Thailand
123
K. Kankam et al.
If t = r = 1 then (1.1) reduces to the split feasibility problem (SFP) studied in [8]. Assume that the MSFP(1.1) is consistent, and let S be the solution set, i.e. S = {x ∗ ∈ C : Ax ∗ ∈ Q} is nonempty. Since its inception in 2005, the multiple-set split feasibility problem (MSFP) can be a unified model for many practical problems such as in signal processing and image reconstruction [18,19], intensity-modulated radiation therapy [3]. Censor et al. [8] proposed the following projection algorithm: xn+1 = P (xn − s∇ p(xn )),
(1.2)
for convenience reasons, they considered the constrained MSFP as find x ∗ ∈ such that x ∗ solves (1.1), where the stepsize s is under the condition that 0 < s L ≤ s ≤ sU < L2 , with L being the Lipschitz constant of ∇ p. However, the computation of the largest eigenvalue (spectral radius) of the matrix A∗ A, which is in general not an easy task in practice, and conservative estimate of the constant L usually results in slow convergence. Many researchers studied the MSFP and introduced various algorithms to solve it such as Masad and Reich [14] gave a note on the multiple-set split convex feasibility problem in Hilbert spaces. Zhang et al. [25] introduced a self-adaptive projection method for solving the multiple-set split feasibility problem. Zhang et al. [26] introduced an efficient simultaneous method for the constrained
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