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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

 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.
Citation: Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control & 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. doi: 10.1090/S0002-9939-05-07719-1. Google Scholar [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. doi: 10.1287/moor.26.2.248.10558. Google Scholar [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. Google Scholar [4] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction,, Inverse Problems, 20 (2004), 103. doi: 10.1088/0266-5611/20/1/006. Google Scholar [5] L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational inequalities,, Appl. Math. Computation, 218 (2011), 1112. Google Scholar [6] L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings,, Taiwanese J. Math., 13 (2009), 1197. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.na.2007.06.040. Google Scholar [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. doi: 10.1016/j.na.2007.09.019. Google Scholar [11] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators,, Optimization, 53 (2004), 475. doi: 10.1080/02331930412331327157. Google Scholar [12] K. Geobel and W. A. Kirk, "Topics in Metric Fixed Point Theory,", Cambridge Studies in Advanced Mathematics, (1990). doi: 10.1017/CBO9780511526152. Google Scholar [13] O. Güler, On the convergence of the proximal point algorithm for convex optimization,, SIAM J. Control Optim., 29 (1991), 403. doi: 10.1137/0329022. Google Scholar [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. doi: 10.1016/S0377-2217(03)00423-5. Google Scholar [15] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems,, Ekonomika i Matematicheskie Metody, 12 (1976), 747. Google Scholar [16] E. S. Levitin and B. T. Polyak, Constrained minimization methods,, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787. Google Scholar [17] G. Marino and H. K. Xu, Convergence of generalized proximal point algorithm,, Comm. Pure Appl. Anal., 3 (2004), 791. doi: 10.3934/cpaa.2004.3.791. Google Scholar [18] C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes,, Nonlinear Anal., 64 (2006), 2400. doi: 10.1016/j.na.2005.08.018. Google Scholar [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. doi: 10.1007/s10957-005-7564-z. Google Scholar [20] Z. Opial, Weak convergence of the sequence of successive approximations of nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 595. doi: 10.1090/S0002-9904-1967-11761-0. Google Scholar [21] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2004.11.017. Google Scholar [24] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings,, J. Optim. Theory Appl., 118 (2003), 417. doi: 10.1023/A:1025407607560. Google Scholar [25] H. K. Xu, Iterative algorithms for nonlinear operators,, J. London Math. Soc., 66 (2002), 240. doi: 10.1112/S0024610702003332. Google Scholar [26] H. K. Xu, Viscosity approximation methods for nonexpansive mappings,, J. Math. Anal. Appl., 298 (2004), 279. doi: 10.1016/j.jmaa.2004.04.059. Google Scholar [27] H. K. Xu, A regularization method for the proximal point algorithm,, J. Global Optim., 36 (2006), 115. doi: 10.1007/s10898-006-9002-7. Google Scholar [28] H. K. Xu, Averaged mappings and the gradient-projection algorithm,, Taiwanese J. Math., (). Google Scholar [29] H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities,, J. Optim. Theory Appl., 119 (2003), 185. doi: 10.1023/B:JOTA.0000005048.79379.b6. Google Scholar [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., (). Google Scholar

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. doi: 10.1090/S0002-9939-05-07719-1. Google Scholar [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. doi: 10.1287/moor.26.2.248.10558. Google Scholar [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. Google Scholar [4] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction,, Inverse Problems, 20 (2004), 103. doi: 10.1088/0266-5611/20/1/006. Google Scholar [5] L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational inequalities,, Appl. Math. Computation, 218 (2011), 1112. Google Scholar [6] L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings,, Taiwanese J. Math., 13 (2009), 1197. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.na.2007.06.040. Google Scholar [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. doi: 10.1016/j.na.2007.09.019. Google Scholar [11] P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators,, Optimization, 53 (2004), 475. doi: 10.1080/02331930412331327157. Google Scholar [12] K. Geobel and W. A. Kirk, "Topics in Metric Fixed Point Theory,", Cambridge Studies in Advanced Mathematics, (1990). doi: 10.1017/CBO9780511526152. Google Scholar [13] O. Güler, On the convergence of the proximal point algorithm for convex optimization,, SIAM J. Control Optim., 29 (1991), 403. doi: 10.1137/0329022. Google Scholar [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. doi: 10.1016/S0377-2217(03)00423-5. Google Scholar [15] G. M. Korpelevich, An extragradient method for finding saddle points and for other problems,, Ekonomika i Matematicheskie Metody, 12 (1976), 747. Google Scholar [16] E. S. Levitin and B. T. Polyak, Constrained minimization methods,, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787. Google Scholar [17] G. Marino and H. K. Xu, Convergence of generalized proximal point algorithm,, Comm. Pure Appl. Anal., 3 (2004), 791. doi: 10.3934/cpaa.2004.3.791. Google Scholar [18] C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes,, Nonlinear Anal., 64 (2006), 2400. doi: 10.1016/j.na.2005.08.018. Google Scholar [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. doi: 10.1007/s10957-005-7564-z. Google Scholar [20] Z. Opial, Weak convergence of the sequence of successive approximations of nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 595. doi: 10.1090/S0002-9904-1967-11761-0. Google Scholar [21] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2004.11.017. Google Scholar [24] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings,, J. Optim. Theory Appl., 118 (2003), 417. doi: 10.1023/A:1025407607560. Google Scholar [25] H. K. Xu, Iterative algorithms for nonlinear operators,, J. London Math. Soc., 66 (2002), 240. doi: 10.1112/S0024610702003332. Google Scholar [26] H. K. Xu, Viscosity approximation methods for nonexpansive mappings,, J. Math. Anal. Appl., 298 (2004), 279. doi: 10.1016/j.jmaa.2004.04.059. Google Scholar [27] H. K. Xu, A regularization method for the proximal point algorithm,, J. Global Optim., 36 (2006), 115. doi: 10.1007/s10898-006-9002-7. Google Scholar [28] H. K. Xu, Averaged mappings and the gradient-projection algorithm,, Taiwanese J. Math., (). Google Scholar [29] H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities,, J. Optim. Theory Appl., 119 (2003), 185. doi: 10.1023/B:JOTA.0000005048.79379.b6. Google Scholar [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., (). Google Scholar
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