Geodesic
Regularized gradient-projection methods for finding the minimum-norm solution of equilibrium and the constrained convex minimization problem
Tian, Ming1 ; Zhang, Hui-Fang2
1 College of Since, Civil Aviation University of China, Tianjin 300300, China;Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China
2 College of Since, Civil Aviation University of China, Tianjin 300300, China
Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 9, p. 5316-5331.

Voir la notice de l'article provenant de la source International Scientific Research Publications

The gradient-projection algorithm (GPA) is an effective method for solving the constrained convex minimization problem. Ordinarily, under some conditions, the minimization problem has more than one solution, so the regulation is used to find the minimum-norm solution of the minimization problem. In this article, we come up with a regularized gradient-projection algorithm to find a common element of the solution set of equilibrium and the solution set of the constrained convex minimization problem, which is the minimum-norm solution of equilibrium and the constrained convex minimization problem. Under some suitable conditions, we can obtain some strong convergence theorems. As an application, we apply our algorithm to solve the split feasibility problem and the constrained convex minimization problem in Hilbert spaces.
DOI : 10.22436/jnsa.009.09.01
Classification : 47H10, 54H25
Keywords: Iterative method, equilibrium problem, constrained convex minimization problem, variational inequality, regularization, minimum-norm.

Tian, Ming 1 ; Zhang, Hui-Fang 2

1 College of Since, Civil Aviation University of China, Tianjin 300300, China;Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China
2 College of Since, Civil Aviation University of China, Tianjin 300300, China 
@article{JNSA_2016_9_9_a0,
     author = {Tian, Ming and Zhang, Hui-Fang},
     title = {Regularized gradient-projection methods for finding the minimum-norm solution of equilibrium and the constrained convex minimization problem},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {5316-5331},
     publisher = {mathdoc},
     volume = {9},
     number = {9},
     year = {2016},
     doi = {10.22436/jnsa.009.09.01},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.09.01/}
}
TY  - JOUR
AU  - Tian, Ming
AU  - Zhang, Hui-Fang
TI  - Regularized gradient-projection methods for finding the minimum-norm solution of equilibrium and the constrained convex minimization problem
JO  - Journal of nonlinear sciences and its applications
PY  - 2016
SP  - 5316
EP  - 5331
VL  - 9
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.09.01/
DO  - 10.22436/jnsa.009.09.01
LA  - en
ID  - JNSA_2016_9_9_a0
ER  - 
%0 Journal Article
%A Tian, Ming
%A Zhang, Hui-Fang
%T Regularized gradient-projection methods for finding the minimum-norm solution of equilibrium and the constrained convex minimization problem
%J Journal of nonlinear sciences and its applications
%D 2016
%P 5316-5331
%V 9
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.09.01/
%R 10.22436/jnsa.009.09.01
%G en
%F JNSA_2016_9_9_a0
Tian, Ming; Zhang, Hui-Fang. Regularized gradient-projection methods for finding the minimum-norm solution of equilibrium and the constrained convex minimization problem. Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 9, p. 5316-5331. doi : 10.22436/jnsa.009.09.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.09.01/

[1] Blum, E.; Oettli, W. From optimization and variational inequalities to equilibrium problems, Math. Student, Volume 63 (1994), pp. 123-145

[2] Brézis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, (French) North-Holland Mathematics Studies, Notas de Matemática, North-Holland Publishing Co., American Elsevier Publishing Co., Inc,, Amsterdam-London-New York, 1973

[3] Byrne, C. A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, Volume 20 (2004), pp. 103-120

[4] Ceng, L. C.; Al-Homidan, S.; Ansari, Q. H.; Yao, J. C. An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Compt. Appl. Math., Volume 223 (2009), pp. 967-974

[5] Ceng, L. C.; Ansari, Q. H.; Wen, C. F. Multi-step implicit iterative methods with regularization for minimization problems and fixed point problems, J. Inequal. Appl., Volume 2013 (2013), pp. 1-26

[6] Ceng, L. C.; Ansari, Q. H.; Yao, J. C. Extragradient-projection method for solving constrained convex minimization problems, Numer. Algebra Control Optim., Volume 1 (2011), pp. 341-359

[7] Ceng, L. C.; Ansari, Q. H.; Yao, J. C. Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Anal., Volume 74 (2011), pp. 5286-5302

[8] Censor, Y.; Elfving, T. A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, Volume 8 (1994), pp. 221-239

[9] Chang, S.-S. Some problems and results in the study of nonlinear analysis, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), Nonlinear Anal., Volume 30 (1997), pp. 4197-4208

[10] Combettes, P. L.; Hirstoaga, S. A. Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., Volume 6 (2005), pp. 117-136

[11] Flam, S. D.; Antipin, A. S. Equilibrium programming using proximal-like algorithms, Math. Programming, Volume 78 (1997), pp. 29-41

[12] He, H.; Liu, S.; Cho, Y. J. An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, J. Comput. Appl. Math., Volume 235 (2011), pp. 4128-4139

[13] Hundal, H. S. An alternating projection that does not converge in norm, Nonlinear Anal., Volume 57 (2004), pp. 35-61

[14] Jung, J. S. Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput., Volume 213 (2009), pp. 498-505

[15] Lin, L. J.; Takahashi, W. A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications, Positivity, Volume 16 (2012), pp. 429-453

[16] Liu, Y. A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., Volume 71 (2009), pp. 4852-4861

[17] Plubtieng, S.; Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., Volume 336 (2007), pp. 455-469

[18] Qin, X.; Cho, Y. J.; Kang, S. M. Convergence analysis on hybrid projection algorithms for equilibrium problems and variational inequality problems, Math. Model. Anal., Volume 14 (2009), pp. 335-351

[19] Takahashi, W. Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama, 2000

[20] Takahashi, S.; Takahashi, W. iscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., Volume 331 (2007), pp. 506-515

[21] Takahashi, S.; Takahashi, W.; Toyoda, M. Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., Volume 147 (2010), pp. 27-41

[22] Tian, M. An application of hybrid steepest descent methods for equilibrium problems and strict pseudocontractions in Hilbert spaces, J. Inequal. Appl., Volume 2011 (2011), pp. 1-15

[23] Tian, M.; Liu, L. Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem, Fixed Point Theory and Appl., Volume 2012 (2012), pp. 1-17

[24] Tian, M.; Liu, L. General iterative methods for equilibrium and constrained convex minimization problem, Optimization, Volume 63 (2014), pp. 1367-1385

[25] Xu, H. K. Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., Volume 150 (2001), pp. 360-378

[26] Xu, H. K. Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., Volume 298 (2004), pp. 279-291

[27] Xu, H. K. Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., Volume 150 (2011), pp. 360-378

[28] Yamada, I. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, Volume 8 (2001), pp. 473-504

[29] Yao, Y. H.; Agarwal, R. P.; Postolache, M.; Liou, Y. C. Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory and Appl., Volume 2014 (2014), pp. 1-14

[30] Yu, Z. T.; Lin, L. J.; Chuang, C. S. A unified study of the split feasible problems with applications, J. Nonlinear Convex Anal., Volume 15 (2014), pp. 605-622

Cité par Sources :