Browse > Article
http://dx.doi.org/10.4134/BKMS.b170868

MODIFIED SUBGRADIENT EXTRAGRADIENT ALGORITHM FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS  

Dang, Van Hieu (Applied Analysis Research Group Faculty of Mathematics and Statistics Ton Duc Thang University)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.5, 2018 , pp. 1503-1521 More about this Journal
Abstract
The paper introduces a modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. Theorem of weak convergence is established under suitable conditions. Several experiments are implemented to illustrate the numerical behavior of the new algorithm and compare it with a well known extragradient method.
Keywords
equilibrium problem; extragradient method; subgradient method; Lipschitz-type condition; pseudomonotone bifunction;
Citations & Related Records
연도 인용수 순위
  • Reference
1 I. Konnov, Equilibrium Models and Variational Inequalities, Mathematics in Science and Engineering, 210, Elsevier B. V., Amsterdam, 2007.
2 G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody 12 (1976), no. 4, 747-756.
3 Yu. V. Malitsky and V. V. Semenov, An extragradient algorithm for monotone variational inequalities, Cybernet. Systems Anal. 50 (2014), no. 2, 271-277.   DOI
4 G. Mastroeni, On auxiliary principle for equilibrium problems, Publicatione del Dipartimento di Mathematica dell, Universita di Pisa, 3 (2000), 1244-1258.
5 G. Mastroeni, Gap functions for equilibrium problems, J. Global Optim. 27 (2003), no. 4, 411-426.   DOI
6 A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom. 15 (1999), no. 1-2, 91-100.
7 L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. 18 (1992), no. 12, 1159-1166.   DOI
8 T. T. V. Nguyen, J. J. Strodiot, and V. H. Nguyen, Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, J. Optim. Theory Appl. 160 (2014), no. 3, 809-831.   DOI
9 P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.
10 J. Contreras, M. Klusch, and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, EEE Trans. Power Syst. 19 (2004), 195-206.   DOI
11 D. V. Hieu, A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space, J. Korean Math. Soc. 52 (2015), no. 2, 373-388.   DOI
12 R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.
13 D. Q. Tran, M. L. Dung, and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization 57 (2008), no. 6, 749-776.   DOI
14 P. T. Vuong, J. J. Strodiot, and V. H. Nguyen, On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space, Optimization (2013). DOI: 10.1080/02331934.2012.759327.   DOI
15 S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), no. 1, Ser. A, 29-41.   DOI
16 K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83, Marcel Dekker, Inc., New York, 1984.
17 D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal. 21 (2016), no. 4, 478-501.   DOI
18 D. V. Hieu, P. K. Anh, and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl. 66 (2017), no. 1, 75-96.   DOI
19 D. V. Hieu, L. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms 73 (2016), no. 1, 197-217.   DOI
20 D. V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 111 (2017), no. 3, 823-840.
21 E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
22 H. Iiduka, Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings, Math. Program. 159 (2016), no. 1-2, Ser. A, 509-538.   DOI
23 A. N. Iusem, G. Kassay, and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program. 116 (2009), no. 1-2, Ser. B, 259-273.   DOI
24 I. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical Systems, 495, Springer-Verlag, Berlin, 2001.
25 I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), 473-504, Stud. Comput. Math., 8, North-Holland, Amsterdam, 2001.
26 P. K. Anh and D. V. Hieu, Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems, Vietnam J. Math. 44 (2016), no. 2, 351-374.   DOI
27 S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
28 Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2011), no. 2, 318-335.   DOI
29 Y. Censor, A. Gibali, and S. Reic, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw. 26 (2011), no. 4-5, 827-845.   DOI
30 Y. Censor, A. Gibali, and S. Reic, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization 61 (2012), no. 9, 1119-1132.   DOI
31 Y. Censor, A. Gibali, and S. Reic, Algorithms for the split variational inequality problem, Numer. Algorithms 59 (2012), no. 2, 301-323.   DOI
32 P. L. Combettes, Quasi-Fejerian analysis of some optimization algorithms, in Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), 115-152, Stud. Comput. Math., 8, North-Holland, Amsterdam, 2001.