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2011, 1(3): 341-359. doi: 10.3934/naco.2011.1.341

Extragradient-projection method for solving constrained convex minimization problems

Luchuan Ceng 1, , Qamrul Hasan Ansari 2, and  Jen-Chih Yao 3,

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

Received  April 2011 Revised  June 2011 Published  September 2011

In this paper, we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a constrained convex minimization problem for a Fr\'{e}chet differentiable function. The iterative process is based on the so-called extragradient-projection method. We derive several weak convergence results for two sequences generated by the proposed iterative process. On the other hand, by applying the viscosity approximation method and the additional projection method (namely, the CQ method) to the extragradient-projection method, respectively, we also provide two modifications of the extragradient-projection method to obtain two strong convergence theorems. The results of this paper represent the supplement, improvement, extension and development of some known results given in the literature.
Keywords: iterative processes., averaged mapping, Extragradient-projection method, constrained convex minimization, nonexpansive mapping, relaxed extragradient-projection method.
Mathematics Subject Classification: Primary: 90C25; Secondary: 65K05, 47H0.
Citation: Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 341-359. doi: 10.3934/naco.2011.1.341
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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