American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity
  • NACO Home
  • This Issue
  • Next Article
    General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity
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 & 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.  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), 248-264. 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), 139-159.  Google Scholar

[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.  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), 1112-1123. Google Scholar

[6]

L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings, Taiwanese J. Math., 13 (2009), 1197-1211.  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), 225-238.  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), 607-632.  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), 1402-1412. 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), 3299-3309. 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), 475-504. doi: 10.1080/02331930412331327157.  Google Scholar

[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.  Google Scholar

[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.  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), 529-544. 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), 747-756. Google Scholar

[16]

E. S. Levitin and B. T. Polyak, Constrained minimization methods, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787-823. Google Scholar

[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.  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), 2400-2411. 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), 191-201. 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), 595-597. 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), 877-898. 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), 189-202.  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), 227-239. 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), 417-428. doi: 10.1023/A:1025407607560.  Google Scholar

[25]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.  Google Scholar

[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.  Google Scholar

[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.  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), 185-201. 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), 1829-1835. 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), 248-264. 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), 139-159.  Google Scholar

[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.  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), 1112-1123. Google Scholar

[6]

L. C. Ceng and S. Huang, Modified extragradient methods for strict pseudo-contractions and monotone mappings, Taiwanese J. Math., 13 (2009), 1197-1211.  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), 225-238.  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), 607-632.  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), 1402-1412. 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), 3299-3309. 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), 475-504. doi: 10.1080/02331930412331327157.  Google Scholar

[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.  Google Scholar

[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.  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), 529-544. 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), 747-756. Google Scholar

[16]

E. S. Levitin and B. T. Polyak, Constrained minimization methods, Zh. vychisl. Mat. mat. Fiz., 6 (1966), 787-823. Google Scholar

[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.  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), 2400-2411. 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), 191-201. 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), 595-597. 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), 877-898. 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), 189-202.  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), 227-239. 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), 417-428. doi: 10.1023/A:1025407607560.  Google Scholar

[25]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.  Google Scholar

[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.  Google Scholar

[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.  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), 185-201. 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

[1]

Biao Qu, Naihua Xiu. A relaxed extragradient-like method for a class of constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 645-654. doi: 10.3934/jimo.2007.3.645
[2]

Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069
[3]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
[4]

Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017
[5]

Abdulkarim Hassan Ibrahim, Jitsupa Deepho, Auwal Bala Abubakar, Kazeem Olalekan Aremu. A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021022
[6]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117
[7]

Janosch Rieger. A learning-enhanced projection method for solving convex feasibility problems. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 555-568. doi: 10.3934/dcdsb.2021054
[8]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045
[9]

Yigui Ou, Wenjie Xu. A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021125
[10]

Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021095
[11]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[12]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[13]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029
[14]

Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen, Yiren Yang. A dimension splitting and characteristic projection method for three-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 127-147. doi: 10.3934/dcdsb.2018111
[15]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
[16]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[17]

Abdulkarim Hassan Ibrahim, Poom Kumam, Min Sun, Parin Chaipunya, Auwal Bala Abubakar. Projection method with inertial step for nonlinear equations: Application to signal recovery. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021173
[18]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178
[19]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035
[20]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial & Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

 Impact Factor: 

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy