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Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy
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 |
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.
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 Google Scholar |
[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 Google Scholar |
[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. |
Initiative point | R-Iter | Iner-R-Iter |
| | |
| | |
| ||
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| | |
| ||
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| | |
|
Initiative point | R-Iter | Iner-R-Iter |
| | |
| | |
| ||
| ||
| | |
| ||
| ||
| | |
|
Initiative point | Iner-KM-R-Iter | |
0.4 | ||
| | |
0.8 | | |
| ||
| 0.4 | |
| | |
0.8 | | |
| ||
| 0.6 | |
| | |
0.8 | | |
|
Initiative point | Iner-KM-R-Iter | |
0.4 | ||
| | |
0.8 | | |
| ||
| 0.4 | |
| | |
0.8 | | |
| ||
| 0.6 | |
| | |
0.8 | | |
|
Initiative point | R-Iter | Iner-R-Iter |
| | |
| | |
| ||
| | |
| | |
| ||
| | |
| | |
Initiative point | R-Iter | Iner-R-Iter |
| | |
| | |
| ||
| | |
| | |
| ||
| | |
| | |
Initiative point | Iner-KM-R-Iter | |
0.6 | ||
| | |
| 0.8 | k=5; s=0.018 |
| ||
| 0.4 | |
| | |
| 0.6 | k=3; s=0.034 |
| ||
| 0.6 | |
| | |
| 0.8 | k= 7; s=0.071 |
|
Initiative point | Iner-KM-R-Iter | |
0.6 | ||
| | |
| 0.8 | k=5; s=0.018 |
| ||
| 0.4 | |
| | |
| 0.6 | k=3; s=0.034 |
| ||
| 0.6 | |
| | |
| 0.8 | k= 7; s=0.071 |
|
R-Iter | Iner-R-Iter | Iner-KM-R-Iter | |
| |||
| |
R-Iter | Iner-R-Iter | Iner-KM-R-Iter | |
| |||
| |
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