Article Contents
Article Contents

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

• * Corresponding author: Meixia Li

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

• 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.

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

 Citation:

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

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

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

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

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 $x^*=(0.744,1.802,-0.223)^T*10^{-5}$ $y^*=(-1.770,7.964,-0.050,-1.210)^T*10^{-5}$ $n=327, s=0.009660$ RTPP $x^*=(0.280,-0.166,0.319)^T$ $y^*=(-0.190,0.477,0.336,0.207)^T$

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 $x^*=(0.285,2.783,-0.0856)^T*10^{-5}$ $y^*=(-2.730,-2.635,-1.595,7.311)^T*10^{-5}$ $n=54878, s=1.100722$ RTPP $x^*=(6.751;-10.660;10.159)^T$ $y^*=(3.244;6.605;5.986;1.070)^T$

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 $x^*=(0.740,4.800,-0.222)^T*10^{-5}$ $y^*=(-0.482,-0.456,-0.289,1.280)^T*10^{-4}$ $n=907710, s=1.785933$ RTPP $x^*=(1.128,-1.722,0.520)^T$ $y^*=(0.096,1.365,3.149,-2.396)^T$

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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