Inertial accelerated algorithms for solving a split feasibility problem

Yazheng Dang; Jie Sun; Honglei Xu

doi:10.3934/jimo.2016078

Article Contents

2017, Volume 13, Issue 3: 1383-1394. Doi: 10.3934/jimo.2016078

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Inertial accelerated algorithms for solving a split feasibility problem

1.
School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
Department of Mathematics and Statistics, Curtin University, Perth, WA 6102, Australia

* Corresponding author:Yazheng Dang. The reviewing process of the paper was handled by Changzhi Wu as a Guest Editor
* Corresponding author:Yazheng Dang. The reviewing process of the paper was handled by Changzhi Wu as a Guest Editor

Received: January 31, 2015

Published: September 2017

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Abstract

Inspired by the inertial proximal algorithms for finding a zero of a maximal monotone operator, in this paper, we propose two inertial accelerated algorithms to solve the split feasibility problem. One is an inertial relaxed-CQ algorithm constructed by applying inertial technique to a relaxed-CQ algorithm, the other is a modified inertial relaxed-CQ algorithm which combines the KM method with the inertial relaxed-CQ algorithm. We prove their asymptotical convergence under some suitable conditions. Numerical results are reported to show the effectiveness of the proposed algorithms.

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Mathematics Subject Classification: Primary: 65K05, 65K10; Secondary: 49J52.

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Table 1. The numerical results of example 5.1

Initiative point	R-Iter	Iner-R-Iter
$ x^{0}=(3.2, 4.2, 5.2)$	$ k=74; s =0.068$	$ k=5; s=0.016$
$ x^{1}=(-0.5843, $	$x^{\ast}=(-0.6200, 1.6180, 1.6216)$	$x^{\ast}=(-1.1281, 1.0720, 1.9694)$
$2.3078, 3.3435) $
$ x^{0}=(10, 0, 10)$	$ k=93; s =0.090$	$ k=84; s=0.085$
$ x^{1}=(2.0825, $	$x^{\ast}=(0.9000, -1.7152, 1.7074)$	$x^{\ast}=(-0.1061, -1.4514, 2.1596)$
$-2.5275, 6.4589) $
$ x^{0}=(2, -5, 2)$	$ k=73; s =0.075$	$ k=35; s =0.035$
$x^{1}=(1.3327, $	$x^{\ast}=(1.1512, -2.7679;1.8616)$	$x^{\ast}=(0.9010, -2.1029, 1.8169)$
$-3.2657, 1.9328) $

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Table 2. The numerical results of example 5.1

Initiative point	$\alpha_{k}$	Iner-KM-R-Iter
$ x^{0}=(3.2, 4.2, 5.2)$	0.4	$ k=3; s=0.016$
$ x^{1}=(-0.5843, 2.3078, 3.3435)$		$x^{\ast}=(-2.6931, 1.2534, 2.2937)$
	0.8	$k=3; s= 0.013$
		$x^{\ast}=(-2.6828, 1.2585, 2.2835)$
$ x^{0}=(10, 0, 10)$	0.4	$ k=76; s =0.086$
$ x^{1}=(2.0825, -2.5275, 6.4589)$		$x^{\ast}=(-0.1346, -2.6392, 2.3046)$
	0.8	$ k= 74; s=0.085 $
		$x^{\ast}=(-0.0799, -2.6190, 2.3611)$
$ x^{0}=(2, -5, 2)$	0.6	$ k=62; s =0.056$
$ x^{1}=(1.3327, -3.2657, 1.9328)$		$x^{\ast}=(0.9006, -2.1031, 1.8171)$
	0.8	$ k=45; s= 0.046$
		$x^{\ast}=(0.9008, -2.1030, 1.8170)$

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Table 3. The numerical results of example 5.2

Initiative point	R-Iter	Iner-R-Iter
$ x^{0}=(0, 0, 0, 0, 0)$	$ k=15$; s $=0.675$	$ k=5$; s $=0.018$
$ x^{1}=(-0.0092, 0, $	$x^{\ast}=(-0.0208, 0, $	$x^{\ast}=(0.0015, 0, $
$-0.0132, -0.0026, -0.0092)$	$-0.0297, -0.0059, -0.0208) $	$-0.0412, -0.0082, -0.0288) $
$ x^{0}=(1, 1, 1, 1, 1)$	$ k=20$; s $=0.083$	$ k=3$; s $=0.0272$
$ x^{1}=(0.3237, 0.5471, $	$x^{\ast}=(0.0171, 0.3822, $	$x^{\ast}=(-0.0784, 0.2935, $
$ 0.2280, 0.4833, 0.3237) $	$ -0.1394, 0.2779, 0.0171) $	$ -0.2378, 0.1873, -0.0784) $
$ x^{0}=(20, 10, 20, 10, 20)$	$ k=22$; s $=0.090$	$ k=6$; s $=0.067$
$x^{1}=(6.1605, 5.0023, $	$x^{\ast}=(0.0837, 0.3910, $	$x^{\ast}=(-0.2490, -0.2117, $
$4.5130, 3.9040, 6.1605)$	$ -0.2155, 0.1915, 0.0837) $	$ -0.1742, -0.1619, -0.2490) $

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Table 4. The numerical results of example 5.2

Initiative point	$\alpha_{k}$	Iner-KM-R-Iter
$ x^{0}=(0, 0, 0, 0, 0)$	0.6	$ k=6$; s $=0.020$
$ x^{1}=(-0.0092, 0, -0.0132, $		$x^{\ast}=(-0.0209, 0, -0.0299, -0.0059, -0.0209)$
$-0.0026, -0.0092) $	0.8	k=5; s=0.018
		$x^{\ast}=(-0.0212, 0, -0.0304, -0.0060, -0.0212)$
$ x^{0}=(1, 1, 1, 1, 1)$	0.4	$ k=3$; s $=0.034$
$ x^{1}=(0.3237, 0.5471, $		$x^{\ast}=(-0.0644, 0.2935, -0.2177, 0.1913, -0.0644)$
$ 0.2280, 0.4833, 0.3237) $	0.6	k=3; s=0.034
		$x^{\ast}=(-0.0691, 0.2935, -0.2244, 0.1899, -0.0691) $
$ x^{0}=(20, 10, 20, 10, 20)$	0.6	$ k=9$; s $=0.072$
$ x^{1}=(6.1605, 5.0023, $		$x^{\ast}=(-0.2263, -0.1045, -0.2337, -0.1094, -0.2263)$
$4.5130, 3.9040, 6.1605) $	0.8	k= 7; s=0.071
		$x^{\ast}=(-0.2283, -0.1610, -0.1881, -0.1342, -0.2283)$

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Table 5. The numerical results of example 5.3

$M, N $	R-Iter	Iner-R-Iter	Iner-KM-R-Iter
$ M=20, N=10$	$ k=436, s =0.970$	$ k=174, s =0.500$	$ k=210, s =0.270$
$M=100, N=90$	$ k=3788, s =0.130$	$ k=602, s =0.680$	$ k=534, s =0.690$

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