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Extragradient-projection method for solving constrained convex minimization problems
1. | Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China |
2. | Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India |
3. | Kaohsiung Medical University, Kaohsiung Medical University, Kaohsiung 80708, Taiwan |
References:
[1] |
H. H. Bauschke, J. V. Burke, F. R. Deutsch, H. S. Hundal and J. D. Vanderwerff, A new proximal point iteration that converges weakly but not in norm, Proc. Amer. Math. Soc., 133 (2005), 1829-1835.
doi: 10.1090/S0002-9939-05-07719-1. |
[2] |
H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces, Math. Operations Research, 26 (2001), 248-264.
doi: 10.1287/moor.26.2.248.10558. |
[3] |
D. P. Bertsekas and E. M. Gafni, Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Program. Study, 17 (1982), 139-159. |
[4] |
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.
doi: 10.1088/0266-5611/20/1/006. |
[5] |
L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational inequalities, Appl. Math. Computation, 218 (2011), 1112-1123. |
[6] |
L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings, Taiwanese J. Math., 13 (2009), 1197-1211. |
[7] |
L. C. Ceng, S. Huang and A. Petrusel, Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings, Taiwanese J. Math., 13 (2009), 225-238. |
[8] |
L. C. Ceng, A. Petrusel, C. Lee and M. M. Wong, Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions, Taiwanese J. Math., 13 (2009), 607-632. |
[9] |
L. C. Ceng, H. K. Xu and J. C. Yao, The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal., 69 (2008), 1402-1412.
doi: 10.1016/j.na.2007.06.040. |
[10] |
L. C. Ceng and J. C. Yao, Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear Anal., 69 (2008), 3299-3309.
doi: 10.1016/j.na.2007.09.019. |
[11] |
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[12] |
K. Geobel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, 1990.
doi: 10.1017/CBO9780511526152. |
[13] |
O. Güler, On the convergence of the proximal point algorithm for convex optimization, SIAM J. Control Optim., 29 (1991), 403-419.
doi: 10.1137/0329022. |
[14] |
D. Han and H. K. Lo, Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities, European J. Operational Research, 159 (2004), 529-544.
doi: 10.1016/S0377-2217(03)00423-5. |
[15] |
G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747-756. |
[16] |
E. S. Levitin and B. T. Polyak, Constrained minimization methods, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787-823. |
[17] |
G. Marino and H. K. Xu, Convergence of generalized proximal point algorithm, Comm. Pure Appl. Anal., 3 (2004), 791-808.
doi: 10.3934/cpaa.2004.3.791. |
[18] |
C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal., 64 (2006), 2400-2411.
doi: 10.1016/j.na.2005.08.018. |
[19] |
N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.
doi: 10.1007/s10957-005-7564-z. |
[20] |
Z. Opial, Weak convergence of the sequence of successive approximations of nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 595-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[21] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[22] |
M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. Ser. A, 87 (2000), 189-202. |
[23] |
T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.
doi: 10.1016/j.jmaa.2004.11.017. |
[24] |
W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
doi: 10.1023/A:1025407607560. |
[25] |
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.
doi: 10.1112/S0024610702003332. |
[26] |
H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.
doi: 10.1016/j.jmaa.2004.04.059. |
[27] |
H. K. Xu, A regularization method for the proximal point algorithm, J. Global Optim., 36 (2006), 115-125.
doi: 10.1007/s10898-006-9002-7. |
[28] |
H. K. Xu, Averaged mappings and the gradient-projection algorithm, Taiwanese J. Math., to appear. |
[29] |
H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl., 119 (2003), 185-201.
doi: 10.1023/B:JOTA.0000005048.79379.b6. |
[30] |
L. C. Zeng, S. Al-Homidan and Q. H. Ansari, Relaxed extragradient-like method for general system of generalized nonlinear mixed equilibrium problems and fixed point problems, Fixed Point Theory Appl., to appear. |
show all references
References:
[1] |
H. H. Bauschke, J. V. Burke, F. R. Deutsch, H. S. Hundal and J. D. Vanderwerff, A new proximal point iteration that converges weakly but not in norm, Proc. Amer. Math. Soc., 133 (2005), 1829-1835.
doi: 10.1090/S0002-9939-05-07719-1. |
[2] |
H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces, Math. Operations Research, 26 (2001), 248-264.
doi: 10.1287/moor.26.2.248.10558. |
[3] |
D. P. Bertsekas and E. M. Gafni, Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Program. Study, 17 (1982), 139-159. |
[4] |
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.
doi: 10.1088/0266-5611/20/1/006. |
[5] |
L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational inequalities, Appl. Math. Computation, 218 (2011), 1112-1123. |
[6] |
L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings, Taiwanese J. Math., 13 (2009), 1197-1211. |
[7] |
L. C. Ceng, S. Huang and A. Petrusel, Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings, Taiwanese J. Math., 13 (2009), 225-238. |
[8] |
L. C. Ceng, A. Petrusel, C. Lee and M. M. Wong, Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions, Taiwanese J. Math., 13 (2009), 607-632. |
[9] |
L. C. Ceng, H. K. Xu and J. C. Yao, The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal., 69 (2008), 1402-1412.
doi: 10.1016/j.na.2007.06.040. |
[10] |
L. C. Ceng and J. C. Yao, Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear Anal., 69 (2008), 3299-3309.
doi: 10.1016/j.na.2007.09.019. |
[11] |
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
doi: 10.1080/02331930412331327157. |
[12] |
K. Geobel and W. A. Kirk, "Topics in Metric Fixed Point Theory," Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, 1990.
doi: 10.1017/CBO9780511526152. |
[13] |
O. Güler, On the convergence of the proximal point algorithm for convex optimization, SIAM J. Control Optim., 29 (1991), 403-419.
doi: 10.1137/0329022. |
[14] |
D. Han and H. K. Lo, Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities, European J. Operational Research, 159 (2004), 529-544.
doi: 10.1016/S0377-2217(03)00423-5. |
[15] |
G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747-756. |
[16] |
E. S. Levitin and B. T. Polyak, Constrained minimization methods, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787-823. |
[17] |
G. Marino and H. K. Xu, Convergence of generalized proximal point algorithm, Comm. Pure Appl. Anal., 3 (2004), 791-808.
doi: 10.3934/cpaa.2004.3.791. |
[18] |
C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal., 64 (2006), 2400-2411.
doi: 10.1016/j.na.2005.08.018. |
[19] |
N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.
doi: 10.1007/s10957-005-7564-z. |
[20] |
Z. Opial, Weak convergence of the sequence of successive approximations of nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 595-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[21] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056. |
[22] |
M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. Ser. A, 87 (2000), 189-202. |
[23] |
T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.
doi: 10.1016/j.jmaa.2004.11.017. |
[24] |
W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
doi: 10.1023/A:1025407607560. |
[25] |
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.
doi: 10.1112/S0024610702003332. |
[26] |
H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.
doi: 10.1016/j.jmaa.2004.04.059. |
[27] |
H. K. Xu, A regularization method for the proximal point algorithm, J. Global Optim., 36 (2006), 115-125.
doi: 10.1007/s10898-006-9002-7. |
[28] |
H. K. Xu, Averaged mappings and the gradient-projection algorithm, Taiwanese J. Math., to appear. |
[29] |
H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theory Appl., 119 (2003), 185-201.
doi: 10.1023/B:JOTA.0000005048.79379.b6. |
[30] |
L. C. Zeng, S. Al-Homidan and Q. H. Ansari, Relaxed extragradient-like method for general system of generalized nonlinear mixed equilibrium problems and fixed point problems, Fixed Point Theory Appl., to appear. |
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