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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 68, 2019 - Issue 11
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Articles

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

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Pages 2099-2124 | Received 14 May 2019, Accepted 09 Aug 2019, Published online: 16 Sep 2019

References

  • Bianchi M, Konnov IV, Pini R. Lexicographic and sequential equilibrium problems. J Global Optim. 2010;46(4):551–560. doi: 10.1007/s10898-009-9439-6
  • Dempe S. Foundations of bilevel programming. Dordrecht: Kluwer Academic Publishers; 2002.
  • Ding XP. Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces. Appl Math Mechan. 2011;32(11):1457–1474. doi: 10.1007/s10483-011-1515-x
  • Moudafi A. Proximal methods for a class of bilevel monotone equilibrium problems. J Global Optim. 2010;47:287–292. doi: 10.1007/s10898-009-9476-1
  • Ceng LC, Latif A, Ansari QH, et al. Hybrid extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl. 2014;2014:222, 1–35. doi: 10.1186/1687-1812-2014-222
  • Ceng LC, Liou YC, Wen CF, et al. Hybrid extragradient viscosity method for general system of variational inequalities. J Inequal Appl. 2015;2015:150, 1–43. doi: 10.1186/s13660-015-0646-z
  • Ceng LC, Liou YC, Wen CF. A hybrid extragradient method for bilevel pseudomonotone variational inequalities with multiple solutions. J Nonlinear Sci Appl. 2016;9:4052–4069. doi: 10.22436/jnsa.009.06.49
  • Ceng LC, Liou YC, Wen CF. Composite relaxed extragradient method for triple hierarchical variational inequalities with constraints of systems of variational inequalities. J Nonlinear Sci Appl. 2017;10:2018–2039. doi: 10.22436/jnsa.010.04.58
  • Ceng LC, Pang CT, Wen CF. Multi-step extragradient method with regularization for triple hierarchical variational inequalities with variational inclusion and split feasibility constraints. J Inequal Appl. 2014;2014:492, 1-40. doi: 10.1186/1029-242X-2014-492
  • Ceng LC, Petrusel A, Yao YC, et al. Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces. Fix Point Theory. 2018;19(2):487–501. doi: 10.24193/fpt-ro.2018.2.39
  • Ceng LC, Sahu DR, Yao JC. A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space. J Inequal Appl. 2014;2014:460, 1–32. doi: 10.1186/1029-242X-2014-460
  • Ceng LC, Wen CF. Relaxed extragradient methods for systems of variational inequalities. J Inequal Appl. 2015;2015:140, 1–41. doi: 10.1186/s13660-015-0648-x
  • Chbani Z, Riahi H. Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities. Optimization. 2015;64:1285–1303. doi: 10.1080/02331934.2013.858397
  • Duc PM, Muu LD. A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization. 2016;65(10):1855–1866. doi: 10.1080/02331934.2016.1195831
  • Bento GC, Cruz Neto JX, Lopes JO, et al. Generalized proximal distances for bilevel equilibrium problems. SIAM J Optim. 2016;26:810–830. doi: 10.1137/140975589
  • Thuy LQ, Hai TN. A projected subgradient algorithm for bilevel equilibrium problems and applications. J Optim Theory Appl. 2017;175(2):411–431. doi: 10.1007/s10957-017-1176-2
  • Anh PN, Kim JK, Muu LD. An extragradient method for solving bilevel variational inequalities. J Global Optim. 2012;52:627–639. doi: 10.1007/s10898-012-9870-y
  • Dempe S, Zemkoho AB. The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Progr. 2013;138:447–473. doi: 10.1007/s10107-011-0508-5
  • Londono G, Lozano A. A bilevel optimization program with equilibrium constraints for an urban network dependent on time. Transp Res Proc. 2014;3:905–914. doi: 10.1016/j.trpro.2014.10.070
  • Santos P, Scheimberg S. An inexact subgradient algorithm for equilibrium problems. Comput Appl Math. 2011;30:91–107.
  • Yao Y, Marino G, Muglia L. A modified Korpelevichś method convergent to the minimum-norm solution of a variational inequality. Optimization. 2014;63:559–569. doi: 10.1080/02331934.2012.674947
  • Anh PN, Thuy LQ, Anh TTH. Strong convergence theorem for the lexicographic Ky Fan inequality. Vietnam J Math. 2018;46(3):517–530. doi: 10.1007/s10013-017-0253-z
  • Cohen G. Auxiliary problem principle and decomposition of optimization problems. J Optim Theory Appl. 1980;32:277–305. doi: 10.1007/BF00934554
  • Konnov IV, Volotskaya EO. Mixed variational inequalities and economic equilibrium problems. J Appl Math. 2002;2(6):289–314. doi: 10.1155/S1110757X02106012
  • Marcotte P. Network design problem with congestion effects: a case of bilevel programming. Math Progr. 1986;34(2):142–162. doi: 10.1007/BF01580580
  • Moudafi A. Proximal point algorithm extended to equilibrium problems. J Nat Geom. 1999;15:91–100.
  • Solodov M. An explicit descent method for bilevel convex optimization. J Conv Anal. 2007;14:227–237.
  • Vuong PT, Strodiot JJ, Nguyen VH. On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization. 2015;64(2):429–451. doi: 10.1080/02331934.2012.759327
  • Mastroeni G. On auxiliary principle for equilibrium problems. In: Daniele P, Giannessi F, Maugeri A, editors. Nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers; 2003.
  • Quoc TD, Muu LD, Nguyen VH. Extragradient algorithms extended to equilibrium problems. Optimization. 2008;57(6):749–776. doi: 10.1080/02331930601122876
  • 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
  • Mastroeni G. Gap functions for equilibrium problems. J Global Optim. 2003;27:411–426. doi: 10.1023/A:1026050425030
  • Bigi G, Castellani M, Pappalardo M. Nonlinear programming techniques for equilibria. Cham: Springer Nature Switzerland; 2019.
  • Maingé PE. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim. 2008;47:1499–1515. doi: 10.1137/060675319
  • Aubin JP, Ekeland I. Applied nonlinear analysis. New York: Wiley; 1984.
  • Anh PN, Le Thi HA. An Armijo-type method for pseudomonotone equilibrium problems and its applications. J Global Optim. 2013;57(3):803–820. doi: 10.1007/s10898-012-9970-8
  • Anh PN. A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization. 2013;62:271–283. doi: 10.1080/02331934.2011.607497
  • Kraikaew R, Saejung S. Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl. 2014;163(2):399–412. doi: 10.1007/s10957-013-0494-2
  • Halpern B. Fixed points of nonexpanding maps. Bull Am Math Soc. 1967;73:957–961. doi: 10.1090/S0002-9904-1967-11864-0
  • Mann WR. Mean value methods in iteration. Proc Am Math Soc. 1953;4:506–510. doi: 10.1090/S0002-9939-1953-0054846-3

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