Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem

Xueling Zhou; Meixia Li; Haitao Che

doi:10.3934/jimo.2020082

Article Contents

2021, Volume 17, Issue 5: 2557-2572. Doi: 10.3934/jimo.2020082

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Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao Shandong, 266590, China
2.
School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China

^* Corresponding author: Meixia Li
^* Corresponding author: Meixia Li

Received: August 2019

Revised: January 2020

Early access: April 2020

Published: September 2021

This project is supported by the Natural Science Foundation of China (Grant No. 11401438, 11571120), Shandong Provincial Natural Science Foundation (Grant No. ZR2017LA002, ZR2019MA022).

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Abstract

The multiple-sets split equality problem is an extended form of the split feasibility problem. It has a wide range of applications in image reconstruction, signal processing, computed tomography, etc. In this paper, we propose a relaxed successive projection algorithm to solve the multiple-sets split equality problem which does not need the prior knowledge of the operator norms, and prove the strong convergence of the algorithm. The numerical examples indicate that the algorithm has good feasibility and effectiveness by comparing with other algorithm.

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Mathematics Subject Classification: Primary: 47H09, 47H05; Secondary: 47J20, 47J25.

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Figure 1. The iteration number of RSPA and RTPP in Case A for Example 4.1

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Figure 2. The iteration number of RSPA and RTPP in Case B for Example 4.1

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Figure 3. The iteration number of RSPA and RTPP in Case C for Example 4.2

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Figure 4. The iteration number of RSPA and RTPP in Case D for Example 4.2

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Table 1. The numerical results of Example 4.1

Init.	$ x_1=(1,1,1)^T $
	$ y_1=(1,1,1,1)^T $
	$ n=14, s=0.003312 $
RSPA	$ x^=(0.744,1.802,-0.223)^T10^{-5} $
	$ y^=(-1.770,7.964,-0.050,-1.210)^T10^{-5} $
	$ n=327, s=0.009660 $
RTPP	$ x^*=(0.280,-0.166,0.319)^T $
	$ y^*=(-0.190,0.477,0.336,0.207)^T $

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Table 2. The numerical results of Example 4.1

Init.	$ x_1=10(1,1,1)^T $
	$ y_1=10(1,1,1,1)^T $
	$ n=30, s=0.005835 $
RSPA	$ x^=(0.285,2.783,-0.0856)^T10^{-5} $
	$ y^=(-2.730,-2.635,-1.595,7.311)^T10^{-5} $
	$ n=54878, s=1.100722 $
RTPP	$ x^*=(6.751;-10.660;10.159)^T $
	$ y^*=(3.244;6.605;5.986;1.070)^T $

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Table 3. The numerical results of Example 4.1

Init.	$ x_1=-10(1,1,1)^T $
	$ y_1=10(1,1,1,1)^T $
	$ n=29, s=0.005965 $
RSPA	$ x^=(0.740,4.800,-0.222)^T10^{-5} $
	$ y^=(-0.482,-0.456,-0.289,1.280)^T10^{-4} $
	$ n=907710, s=1.785933 $
RTPP	$ x^*=(1.128,-1.722,0.520)^T $
	$ y^*=(0.096,1.365,3.149,-2.396)^T $

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Table 4. The numerical results of Example 4.2

				RSPA		RTPP
	$ J $	$ N $	$ M $	$ n $	$ s $	$ n $	$ s $
	$ 10 $	$ 20 $	$ 30 $	$ 13 $	$ 0.001976 $	1370	0.078588
Case 1	40	30	40	14	0.002048	20842	3.989155
	60	60	60	15	0.002840	24600	10.765349
	$ 10 $	$ 20 $	$ 30 $	15	0.001523	9573	0.669758
Case 2	40	30	40	17	0.002967	21674	4.326832
	60	60	60	18	0.003256	23970	12.725284
	$ 10 $	$ 20 $	$ 30 $	16	0.001644	1338	0.078992
Case 3	40	30	40	17	0.001897	21237	4.291747
	60	60	60	18	0.003552	24110	10.261271
	$ 10 $	$ 20 $	$ 30 $	15	0.001891	9573	0.528336
Case 4	40	30	40	17	0.002379	21674	4.199953
	60	60	60	18	0.002865	23970	10.365368

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