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July  2017, 13(3): 1383-1394. doi: 10.3934/jimo.2016078

## 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

Received  February 2015 Published  October 2016

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.

Citation: Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078
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##### References:
The numerical results of example 5.1
 Initiative point R-Iter Iner-R-Iter $x^{0}=(3.2, 4.2, 5.2)$ $k=74; s =0.068$ $k=5; s=0.016$ $x^{1}=(-0.5843,$ $x^{\ast}=(-0.6200, 1.6180, 1.6216)$ $x^{\ast}=(-1.1281, 1.0720, 1.9694)$ $2.3078, 3.3435)$ $x^{0}=(10, 0, 10)$ $k=93; s =0.090$ $k=84; s=0.085$ $x^{1}=(2.0825,$ $x^{\ast}=(0.9000, -1.7152, 1.7074)$ $x^{\ast}=(-0.1061, -1.4514, 2.1596)$ $-2.5275, 6.4589)$ $x^{0}=(2, -5, 2)$ $k=73; s =0.075$ $k=35; s =0.035$ $x^{1}=(1.3327,$ $x^{\ast}=(1.1512, -2.7679;1.8616)$ $x^{\ast}=(0.9010, -2.1029, 1.8169)$ $-3.2657, 1.9328)$
 Initiative point R-Iter Iner-R-Iter $x^{0}=(3.2, 4.2, 5.2)$ $k=74; s =0.068$ $k=5; s=0.016$ $x^{1}=(-0.5843,$ $x^{\ast}=(-0.6200, 1.6180, 1.6216)$ $x^{\ast}=(-1.1281, 1.0720, 1.9694)$ $2.3078, 3.3435)$ $x^{0}=(10, 0, 10)$ $k=93; s =0.090$ $k=84; s=0.085$ $x^{1}=(2.0825,$ $x^{\ast}=(0.9000, -1.7152, 1.7074)$ $x^{\ast}=(-0.1061, -1.4514, 2.1596)$ $-2.5275, 6.4589)$ $x^{0}=(2, -5, 2)$ $k=73; s =0.075$ $k=35; s =0.035$ $x^{1}=(1.3327,$ $x^{\ast}=(1.1512, -2.7679;1.8616)$ $x^{\ast}=(0.9010, -2.1029, 1.8169)$ $-3.2657, 1.9328)$
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 $k=45; s= 0.046$ $x^{\ast}=(0.9008, -2.1030, 1.8170)$
 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 $k=45; s= 0.046$ $x^{\ast}=(0.9008, -2.1030, 1.8170)$
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 $=0.067$ $x^{1}=(6.1605, 5.0023,$ $x^{\ast}=(0.0837, 0.3910,$ $x^{\ast}=(-0.2490, -0.2117,$ $4.5130, 3.9040, 6.1605)$ $-0.2155, 0.1915, 0.0837)$ $-0.1742, -0.1619, -0.2490)$
 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 $=0.067$ $x^{1}=(6.1605, 5.0023,$ $x^{\ast}=(0.0837, 0.3910,$ $x^{\ast}=(-0.2490, -0.2117,$ $4.5130, 3.9040, 6.1605)$ $-0.2155, 0.1915, 0.0837)$ $-0.1742, -0.1619, -0.2490)$
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 $x^{\ast}=(-0.2283, -0.1610, -0.1881, -0.1342, -0.2283)$
 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 $x^{\ast}=(-0.2283, -0.1610, -0.1881, -0.1342, -0.2283)$
The numerical results of example 5.3
 $M, N$ R-Iter Iner-R-Iter Iner-KM-R-Iter $M=20, N=10$ $k=436, s =0.970$ $k=174, s =0.500$ $k=210, s =0.270$ $M=100, N=90$ $k=3788, s =0.130$ $k=602, s =0.680$ $k=534, s =0.690$
 $M, N$ 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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