A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem

Mohammad Eslamian; Ahmad Kamandi

doi:10.3934/jimo.2021210

Article Contents

2023, Volume 19, Issue 2: 868-889. Doi: 10.3934/jimo.2021210

This issue Previous Article On best linear unbiased estimation and prediction under a constrained linear random-effects model Next Article A new hybrid

$ l_p $

$ l_2 $

model for sparse solutions with applications to image processing

A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem

Mohammad Eslamian^1,2, , and
Ahmad Kamandi^1,

1.
Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
2.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

^* Corresponding author: Mohammad Eslamian
^* Corresponding author: Mohammad Eslamian

Received: July 2021

Revised: October 2021

Early access: December 2021

Published: February 2023

This research was in part supported by a grant from IPM (No.1400470032).

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Abstract

In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view.

Keywords:

Mathematics Subject Classification: Primary: 47H05, 47H10; Secondary: 65Y05.

Citation:

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Figure 1. The graph of the error $ \|x_n-x_{n-1}\|_2 $

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Figure 2. The graph of $ x_n $

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Figure 3. The graph of the value of $ \frac{1}{2} \|Tx_n-b\|^2 $ for the algorithms

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Figure 4. Comparison of the new algorithm and the Algorithm 1 to recovery of a sparse signal with 5% nonzero elements

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Table 1. The results of the new algorithm for Example 5.1

Starting points		CPU time	No. iterations
$ x_0=t^2 $	$ x_1=sin(2t) $	0.62 s	1
$ x_0=t $	$ x_1=3e^{-2t} $	1.42 s	2
$ x_0=10\;\; t\;\; sin(5t) $	$ x_1=10\;\; t\;\; sin(5t) $	0.61 s	1

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Table 2. Numerical results of comparison of the new algorithm and the Algorithm 1 for Example 5.2

Starting point(s)	Algorithm 1		The new algorithm
	No. iterations	CPU time	No. iterations	CPU time
$ [-5,4,-3,2,-1] $	63	0.0133	31	0.0025
$ [-50,40,-30,20,-1] $	72	0.0263	35	0.0093
$ [5,4,3,2,1] $	64	0.0146	31	0.0120
$ [50,40,30,20,10] $	72	0.0150	29	0.0095

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Table 3. The details of iterations of the new algorithm for Example 5.2

n	$ x_n $	$ \\|x_n-x_{n-1}\\|_2 $
0	[-5.000, 4.0000, -3.0000, 2.0000, -1.0000]	—
1	[-5.000, 4.0000, -3.0000, 2.0000, -1.0000]	—
2	[-6.3896, 5.0620, -4.2310, 2.4069, -1.5759]	2.2520e+00
8	[-0.0295, 0.0190, -0.0547, -0.0021, -0.0336]	1.8203e-01
14	[ 0.0019, -0.0014, 0.0024, -0.0003, 0.0013]	2.6378e-03
20	[ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000]	5.5509e-05
22	[ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000]	8.9023e-06

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Table 4. Numerical results of comparison of the new algorithm and The Algorithm 1 to recovery of a sparse signal with $ p $ % nonzero elements

p	Algorithm 1		New Algorithm
	CPU time	No. iterations	CPU time	No. iterations
2 %	1.8809 s	606	0.3698 s	129
5 %	2.2180 s	790	0.4061 s	142
10 %	1.1052 s	374	0.5425 s	174

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