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

Inertial accelerated algorithms for solving a split feasibility problem

Yazheng Dang 1,, , Jie Sun 2, and  Honglei Xu 2,

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.

Keywords: Asymptotical convergence, inertial technique, nonsmooth analysis, split feasibility problem, subdifferential.
Mathematics Subject Classification: Primary: 65K05, 65K10; Secondary: 49J52.
Citation: Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078
References:
[1]

F. Alvarez, On the minimizing property of a second order dissipative dynamical system in Hilbert spaces, SIAM Journal on Control and Optimization, 39 (2000), 1102-1119.  doi: 10.1137/S0363012998335802.

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via Discretization of a nonlinear oscillator with damping, Set-Valued Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[3]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM Journal on Optimization, 14 (2003), 773-782.  doi: 10.1137/S1052623403427859.

[4]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[5]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.

[6]

C. Byrne, An unified treatment of some iterative algorithm algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[7]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS Journal on Computing, 16 (2004), 255-265.  doi: 10.1287/ijoc.1030.0046.

[8]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Mathematical Programming, 42 (1998), 307-325.  doi: 10.1007/BF01589408.

[9]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[10]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets solit feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[11]

G. Crombez, A geometrical look at iterative methods for operators with fixed points, Numerical Functional Analysis and Optimization, 26 (2005), 137-175.  doi: 10.1081/NFA-200063882.

[12]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.

[13]

F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications, 105-121, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 356, Kluwer Acad. Publ. , Dordrecht, 1992

[14]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-Like algorithm for split feasibility problem, Inverse Problems 27 (2011), 015007, 9 pp. doi: 10.1088/0266-5611/27/1/015007.

[15]

M. Fukushima, On the convergence of a class of outer approximation algorithms for convex programs, Journal of Computational and Applied Mathematics, 10 (1984), 147-156.  doi: 10.1016/0377-0427(84)90051-7.

[16]

M. Fukushima, A relaxed projection method for variational inequalities, Mathematics Programming, 35 (1986), 58-70.  doi: 10.1007/BF01589441.

[17]

Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. 

[18]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[19]

P. E. Mainge, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set-valued Analysis, 15 (2007), 67-79.  doi: 10.1007/s11228-006-0027-3.

[20]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms, Journal of Computational and Applied Mathematics, 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[21]

Z. Opial, Weak convergence of the sequence of successive approximations for non-expansive mappings, Bull. American Mathematical Society, 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.

[22]

B. Qu and N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra and Its Application, 428 (2008), 1218-1229.  doi: 10.1016/j.laa.2007.03.002.

[23]

H. Xu, A variabe Krasnoselski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.

[24]

Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[25]

A. L. Yan, G. Y. Wang and N. H. Xiu, Robust solutions of split feasibility problem with uncertain linear operator, Journal of Industrial and Management Optimization, 3 (2007), 749-761.  doi: 10.3934/jimo.2007.3.749.

[26]

J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799.  doi: 10.1088/0266-5611/21/5/017.

show all references

References:
[1]

F. Alvarez, On the minimizing property of a second order dissipative dynamical system in Hilbert spaces, SIAM Journal on Control and Optimization, 39 (2000), 1102-1119.  doi: 10.1137/S0363012998335802.

[2]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via Discretization of a nonlinear oscillator with damping, Set-Valued Analysis, 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[3]

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM Journal on Optimization, 14 (2003), 773-782.  doi: 10.1137/S1052623403427859.

[4]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[5]

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.

[6]

C. Byrne, An unified treatment of some iterative algorithm algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.  doi: 10.1088/0266-5611/20/1/006.

[7]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS Journal on Computing, 16 (2004), 255-265.  doi: 10.1287/ijoc.1030.0046.

[8]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Mathematical Programming, 42 (1998), 307-325.  doi: 10.1007/BF01589408.

[9]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[10]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets solit feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[11]

G. Crombez, A geometrical look at iterative methods for operators with fixed points, Numerical Functional Analysis and Optimization, 26 (2005), 137-175.  doi: 10.1081/NFA-200063882.

[12]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.

[13]

F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications, 105-121, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 356, Kluwer Acad. Publ. , Dordrecht, 1992

[14]

Y. Dang and Y. Gao, The strong convergence of a KM-CQ-Like algorithm for split feasibility problem, Inverse Problems 27 (2011), 015007, 9 pp. doi: 10.1088/0266-5611/27/1/015007.

[15]

M. Fukushima, On the convergence of a class of outer approximation algorithms for convex programs, Journal of Computational and Applied Mathematics, 10 (1984), 147-156.  doi: 10.1016/0377-0427(84)90051-7.

[16]

M. Fukushima, A relaxed projection method for variational inequalities, Mathematics Programming, 35 (1986), 58-70.  doi: 10.1007/BF01589441.

[17]

Y. Gao, Piecewise smooth Lyapunov function for a nonlinear dynamical system, Journal of Convex Analysis, 19 (2012), 1009-1015. 

[18]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[19]

P. E. Mainge, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set-valued Analysis, 15 (2007), 67-79.  doi: 10.1007/s11228-006-0027-3.

[20]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms, Journal of Computational and Applied Mathematics, 219 (2008), 223-236.  doi: 10.1016/j.cam.2007.07.021.

[21]

Z. Opial, Weak convergence of the sequence of successive approximations for non-expansive mappings, Bull. American Mathematical Society, 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.

[22]

B. Qu and N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra and Its Application, 428 (2008), 1218-1229.  doi: 10.1016/j.laa.2007.03.002.

[23]

H. Xu, A variabe Krasnoselski-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.

[24]

Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.  doi: 10.1088/0266-5611/20/4/014.

[25]

A. L. Yan, G. Y. Wang and N. H. Xiu, Robust solutions of split feasibility problem with uncertain linear operator, Journal of Industrial and Management Optimization, 3 (2007), 749-761.  doi: 10.3934/jimo.2007.3.749.

[26]

J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791-1799.  doi: 10.1088/0266-5611/21/5/017.

Table 1.  The numerical results of example 5.1
Initiative pointR-IterIner-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$$ k=35; s =0.035$
$x^{1}=(1.3327, $ $x^{\ast}=(1.1512, -2.7679;1.8616)$ $x^{\ast}=(0.9010, -2.1029, 1.8169)$
$-3.2657, 1.9328) $
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Initiative pointR-IterIner-R-Iter
$ x^{0}=(3.2, 4.2, 5.2)$ $ k=74; s =0.068$$ 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) $
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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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$
$x^{\ast}=(0.9008, -2.1030, 1.8170)$
Table 3.  The numerical results of example 5.2
Initiative pointR-IterIner-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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Initiative pointR-IterIner-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) $
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) $0.8k=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) $0.6k=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) $0.8k= 7; s=0.071
$x^{\ast}=(-0.2283, -0.1610, -0.1881, -0.1342, -0.2283)$
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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) $0.8k=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) $0.6k=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) $0.8k= 7; s=0.071
$x^{\ast}=(-0.2283, -0.1610, -0.1881, -0.1342, -0.2283)$
Table 5.  The numerical results of example 5.3
$M, N $R-IterIner-R-IterIner-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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$M, N $R-IterIner-R-IterIner-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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