Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

Guash Haile Taddele; Poom Kumam; Habib ur Rehman; Anteneh Getachew Gebrie

doi:10.3934/jimo.2021172

Article Contents

2023, Volume 19, Issue 1: 1-29. Doi: 10.3934/jimo.2021172

This issue Previous Article Next Article Projection method with inertial step for nonlinear equations: Application to signal recovery

Self adaptive inertial relaxed $ CQ $ algorithms for solving split feasibility problem with multiple output sets

1.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology, Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
3.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics, College of Computational and Natural Science, Debre Berhan University, Debre Berhan, PO. Box 445, Ethiopia

^* Corresponding author: E-mail address: poom.kum@kmutt.ac.th (Poom Kumam)
^* Corresponding author: E-mail address: poom.kum@kmutt.ac.th (Poom Kumam)

Received: March 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: January 2023

The first author is supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi grant No.37/2561

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Abstract

In this paper, we propose two new self-adaptive inertial relaxed $ CQ $ algorithms for solving the split feasibility problem with multiple output sets in the framework of real Hilbert spaces. The proposed algorithms involve computing projections onto half-spaces instead of onto the closed convex sets, and the advantage of the self-adaptive step size introduced in our algorithms is that it does not require the computation of operator norm. We establish and prove weak and strong convergence theorems for the iterative sequences generated by the introduced algorithms for solving the aforementioned problem. Moreover, we apply the new results to solve some other problems. Finally, we present some numerical examples to illustrate the implementation of our algorithms and compared them to some existing results.

Keywords:

Split feasibility problem,
split feasibility problem with multiple output sets,
self-adaptive technique,
inertial technique,
half-space.

Mathematics Subject Classification: 47H09, 65J15, 65K05, 65K10, 49J52.

Citation:

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Figure 1. Comparison of Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $ \epsilon $

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Figure 2. Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17) for different choices of initial points

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Table 1. Algorithm 1 and Algorithm 3 for $ \epsilon = 10^{-6} $ and different choices of $ \rho_{1}^{n}, \rho_{2}^{n} $ and $ \theta $

	$ \rho_{1}^{n}=\frac{3n}{4n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{n}{2n+1}=\rho_{2}^{n} $			$ \rho_{1}^{n}=\frac{3n}{20n+1}=\rho_{2}^{n} $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	22	0.018314	8.41E-07	32	0.025566	7.17E-07	53	0.024086	9.40E-07	74	0.02651	9.69E-07
Algorithm 3	83	0.023672	9.67E-07	91	0.042918	9.61E-07	157	0.028161	9.88E-07	207	0.034091	9.80E-07

	$ \theta=0 $			$ \theta=0.15 $			$ \theta=0.25 $			$ \theta=0.5 $
	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En	Iter.(n)	CPU(s)	En
Algorithm 1	45	0.025492	8.00E-07	57	0.02797	7.78E-07	43	0.029967	9.76E-07	51	0.02619	8.02E-07
Algorithm 3	111	0.026377	9.79E-07	136	0.030002	9.82E-07	91	0.026963	9.81E-07	84	0.024384	9.82E-07

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Table 2. Algorithm 1, Algorithm 3, Scheme (16), Scheme (17) and Scheme (5.1) for different choices of $ \epsilon $

		Algorithm 1	Algorithm 3	Scheme (16)	Scheme (17)	Scheme (5.1)
$ \epsilon=10^{-6} $	Iter.(n)	24	75	180	111	75
	CPU(s)	0.01667	0.02255	0.023436	0.037266	0.029002
	$ E_n $	8.25E-07	9.67E-07	9.74E-07	9.82E-07	9.92E-07
$ \epsilon=10^{-7} $	Iter.(n)	30	134	174	282	211
	CPU(s)	0.01962	0.025028	0.025425	0.026567	0.033771
	$ E_n $	6.17E-08	9.83E-08	9.80E-08	9.96E-08	9.99E-08
$ \epsilon=10^{-8} $	Iter.(n)	41	276	470	537	770
	CPU(s)	0.024448	0.029783	0.033593	0.035215	0.038546
	$ E_n $	8.68E-09	9.95E-09	9.84E-09	9.99E-09	9.98E-09
$ \epsilon=10^{-9} $	Iter.(n)	49	479	496	2024	2263
	CPU(s)	0.026591	0.037697	0.036028	0.039359	0.088713
	$ E_n $	6.98E-10	9.93E-10	9.83E-10	1.00E-09	1.00E-09

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Table 3. Comparison of Algorithm 5, Scheme (13), Scheme (14) and Scheme (17)

Algorithm 5			Scheme (14)			Scheme (13)			Scheme (17)
Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)	Iter.(n)	$ E_n $	CPU(s)
1	1149.360361		1	231.8966545		1	199.2220601		1	640.4875017
2	28.03259245		2	91.40575598		2	69.5163167		2	12.35158962
3	0.811105412		3	32.14832803		3	20.87177641		3	2.434404763
4	0.202894757		4	10.25120406		4	6.548286817		4	0.513146017
5	0.050765569		5	3.137538418		5	2.301982084		5	0.140142486
6	0.012707645		6	1.056033824		6	0.966763557		6	0.083619119
7	0.003183583		7	0.449136255		7	0.488009333		7	0.059127765
8	0.000798736		8	0.228594093		8	0.279757135		8	0.041809171
9	0.000200925		9	0.11848334		9	0.170839653		9	0.029562999
10	5.07839E-05	69.73042	10	0.056872251		10	0.106846934		10	0.020903734
			11	0.024488093		11	0.06722031		11	0.014780829
			12	0.009391695		12	0.042254471		12	0.010451389
			13	0.00322014		13	0.026486762		13	0.007390099
			14	0.001009238		14	0.01655399		14	0.005225502
			15	0.000311098		15	0.010319917		15	0.003694944
			16	0.000109788		16	0.006420625		16	0.002612704
			17	4.93538E-05	140.9432	17	0.003988518		17	0.001847463
						18	0.002474828		18	0.001306366
						19	0.001534285		19	0.000923758
						20	0.000950584		20	0.000653215
						21	0.000588668		21	0.000461913
						22	0.000364417		22	0.00032664
						23	0.000225534		23	0.000230986
						24	0.000139554		24	0.000163347
						25	8.6339E-05	173.0115	25	0.000115517
									26	8.16939E-05	132.0709

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