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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 68, 2019 - Issue 8
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Articles

A new projection algorithm for solving constrained equilibrium problems in Hilbert spaces

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Pages 1447-1470 | Received 25 Dec 2016, Accepted 14 Mar 2019, Published online: 18 Apr 2019

References

  • Santos PSM, Scheimberg S. A modified projection algorithm for constrained equilibrium problems. Optimization. 2017;66:2051–2062. doi: 10.1080/02331934.2016.1182528
  • Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Student. 1994;63:123–146.
  • Chamnarnpan T, Phiangsungnoen S, Kumam P. A new hybrid extragradient algorithm for solving the equilibrium and variational inequality problems. Afr Mat. 2015;26:87–98. doi: 10.1007/s13370-013-0187-x
  • Cholamjiak P, Suantai S. Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups. J Global Optim. 2013;57:1277–1297. doi: 10.1007/s10898-012-0029-7
  • Combettes PL, Hirstoaga SA. Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal. 2005;6:117–136.
  • Dong NTP, Strodiot JJ, Van NTT, et al. A family of extragradient methods for solving equilibrium problems. J Ind Manag Optim. 2015;11:619–630.
  • Khatibzadeh H, Mohebbi V. Proximal point algorithm for infinite pseudo-monotone bifunctions. Optimization. 2016;65:1629–1639. doi: 10.1080/02331934.2016.1153639
  • Plubtieng S, Punpaeng R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl. 2007;336:455–469. doi: 10.1016/j.jmaa.2007.02.044
  • Quoc TD, Muu LD, Hien NV. Extragradient algorithms extended to equilibrium problems. Optimization. 2008;57:749–776. doi: 10.1080/02331930601122876
  • Strodiot JJ, Vuong PT, Van NTT. A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J Global Optim. 2016;64:159–178. doi: 10.1007/s10898-015-0365-5
  • Suantai S, Cholamjiak P. Algorithms for solving generalized equilibrium problems and fixed points of nonexpansive semigroups in Hilbert spaces. Optimization. 2014;63:799–815. doi: 10.1080/02331934.2012.684355
  • Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008;69:1025–1033. doi: 10.1016/j.na.2008.02.042
  • Vuong PT, Strodiot JJ, Hien NV. On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization. 2015;64:429–451. doi: 10.1080/02331934.2012.759327
  • Vuong PT, Strodiot JJ, Hien NV. Extragradient methods and linearsearch algorithms for solving Ky Fan inequalities and fixed point problems. J Optim Theory Appl. 2012;155:605–627. doi: 10.1007/s10957-012-0085-7
  • Censor Y, Gibali A, Reich S. Algorithms for the split variational inequality problem. Numer Algorithms. 2012;59:301–323. doi: 10.1007/s11075-011-9490-5
  • He Z. The split equilibrium problem and its convergence algorithms. J Inequal Appl. 2012;2012:162, 15 pp, doi: 10.1186/1029-242X-2012-162
  • Byrne C. Iterative optimization in inverse problems. Boca Raton (FL): CCR Press; 2013. (Monograph and research notes in mathematics). ISBN 1482222337.
  • Cholamjiak W, Cholamjiak P, Suantai S. An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J Fixed Point Theory Appl. 2018;20:42. Available from: https://doi.org/10.1007/s11784-018-0526-5.
  • Suantai S, Pholasa N, Cholamjiak P. Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. RACSAM. 2018. Available from: https://doi.org/10.1007/s13398-018-0535-7.
  • Suantai S, Pholasa N, Cholamjiak P. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J Indust Manag Optim. 2018;14:1595–1615.
  • Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335. doi: 10.1007/s10957-010-9757-3
  • Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Meth Softw. 2011;26:827–845. doi: 10.1080/10556788.2010.551536
  • Crouzeix JP, Marcotte P, Zhu D. Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math Program Ser A. 2000;88:521–539. doi: 10.1007/PL00011384
  • Hadjisavvas N, Schaible S. Pseudomonotone∗ maps and cutting plane property. J Global Optim. 2009;43:565–575. doi: 10.1007/s10898-008-9335-5
  • Anh PN, An LTH. The subgradient extragradient method extended to equilibrium problems. Optimization. 2015;64:225–248. doi: 10.1080/02331934.2012.745528
  • Chbani Z, Riahi H. Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities. Optim Lett. 2013;7:185–206. doi: 10.1007/s11590-011-0407-y
  • Moreau JJ. Fonctions convexes duales et points proximaux dans un espace hilbertien. C R Acad Sci Paris. 1962;255:2897–2899.
  • Minty GJ. Monotone (nonlinear) operators in Hilbert space. Duke Math J. 1962;29:341–346. doi: 10.1215/S0012-7094-62-02933-2
  • Karamardian S. Complementarity problems over cones with monotone and pseudomonotone maps. J Optim Theory Appl. 1976;18:445–454. doi: 10.1007/BF00932654
  • Dinh BV, Muu LD. A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria. Optimization. 2015;64:559–575. doi: 10.1080/02331934.2013.864290
  • Santos PSM, Scheimberg S. An inexact subgradient algorithm for equilibrium problems. Comput Appl Math. 2011;30:91–107.
  • Tan KK, Xu HK. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J Math Anal Appl. 1993;178:301–308. doi: 10.1006/jmaa.1993.1309
  • Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. New York: Springer; 2011.
  • Solodov MV, Svaiter BF. A new projection method for variational inequality problems. SIAM J Control Optim. 1999;37:765–776. doi: 10.1137/S0363012997317475
  • Thuy LQ, Hai TN. A projected subgradient algorithm for bilevel equilibrium problems and applications. J Optim Theory Appl. 2017;175:411–431. doi: 10.1007/s10957-017-1176-2

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