Nonlinear iteration method for proximal split feasibility problems

Abstract

Communicated by: R. Picard MOS Classification: 47H06; 47H09; 47J05; 47J25The purpose of this paper is to introduce iterative algorithm which is a combination of hybrid viscosity approximation method and the hybrid steepest-descent method for solving proximal split feasibility problems and obtain the strong convergence of the sequences generated by the iterative scheme under certain weaker conditions in Hilbert spaces. Our results improve many recent results on the topic in the literature. Several numerical experiments are presented to illustrate the effectiveness of our proposed algorithm, and these numerical results show that our result is computationally easier and faster than previously known results on proximal split feasibility problem. KEYWORDSHilbert spaces, Moreau-Yosida approximate, proximal split feasibility problems, strong convergencewhere H 1 andH 2 are 2 real Hilbert spaces, f ∶ H 1 → R∪{+∞}, g ∶ H 2 → R∪{+∞} 2 proper, convex, lower semicontinuous functions, and A ∶ H 1 → H 2 a bounded linear operator; g (y) = min u∈H 2 {g(u) + 1 2 ||u − y|| 2 } stands for the Moreau-Yosida approximate of the function g of parameter .

Strong convergence of an inertial extrapolation method for a split system of minimization problems

“…In this article, we introduce a strong convergence theorem for an inertial extrapolation-type algorithm for solving a SSMP (7). The problem we considered in this article is general for many of the problems considered in the literature concerning approximation of an unconstrained minimization problem, see for example [25][26][27][28]24,23]. Our result can also be applied to find a solution of the split system of inclusion problem, the MSSFP, and the split system of equilibrium problem.…”

“…is a solution of SMP (4) and the iterative process stops; otherwise, we set ≔ + n n 1 and go to (5). Based on Moudafi and Thakur [21] many iterative algorithms are proposed for solving SMP (4), see for example those by Abbas et al in [23], Shehu et al in [24], Shehu and Iyiola in [25][26][27][28] and Shehu and Ogbuisi in [29].…”

“…Note that if = f f i for all ∈ i Φ and = g g j for all ∈ j Ψ, then problem (7) reduces to the SMP (4) that is the problem considered in [21,[23][24][25][26][27][28][29]. The aims of this study are twofold: to improve the weak convergence result of an inertial extrapolation-type algorithm proposed by Shehu and Iyiola [25] to a strong convergence result for an approximation of a solution of the SMP (4), and to accelerate and improve the results in [9,10] in solving the SSMP (7).…”

New inertial method for generalized split variational inclusion problems

“…(c) Our results can be modified for solving proximal split feasibility problem ( [27,31,33,44,45,46]): find x ∈ H 1 such that x ∈ argmin f and Ax ∈ argmin g, where f : H 1 → R ∪ {+∞} and g : H 2 → R ∪ {+∞} are proper, convex and lower semi-continuous functionals. In this case, we take s = 1 = t and B 1 = ∂f, D 1 = ∂g.…”