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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 69, 2020 - Issue 5
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Articles

Ekeland variational principles for set-valued functions with set perturbations

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Pages 925-960 | Received 23 Aug 2018, Accepted 15 Jul 2019, Published online: 13 Aug 2019

References

  • Ekeland I. Sur les probèmes variationnels. C R Acad Sci Paris. 1972;275:1057–1059.
  • Ekeland I. On the variational principle. J Math Anal Appl. 1974;47:324–353. doi: 10.1016/0022-247X(74)90025-0
  • Chen GY, Huang XX, Yang XG. Vector optimization: set-valued and variational analysis. Berlin: Springer-Verlag; 2005.
  • Ekeland I. Nonconvex minimization problems. Bull Amer Math Soc (NS). 1979;1:443–474. doi: 10.1090/S0273-0979-1979-14595-6
  • Göpfert A, Riahi H, Tammer Chr, et al. Variational methods in partially ordered spaces. New York: Springer-Verlag; 2003.
  • Khan AA, Tammer C, Zălinescu C. Set-valued optimization: an introduction with applications. Berlin Heidelberg: Springer-Verlag; 2015.
  • Araya Y. Ekeland's variational principle and its equivalent theorems in vector optimization. J Math Anal Appl. 2008;346:9–16. doi: 10.1016/j.jmaa.2008.04.055
  • Bao TQ, Mordukhovich BS. Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 2007;36:531–562.
  • Bao TQ, Mordukhovich BS. Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math Program Ser A. 2010;122:301–347. doi: 10.1007/s10107-008-0249-2
  • Bednarczuk EM, Przybyla MJ. The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors. SIAM J Optim. 2007;18:907–913. doi: 10.1137/060658989
  • Bednarczk EM, Zagrodny D. Vector variational principle. Arch Math (Basel). 2009;93:577–586. doi: 10.1007/s00013-009-0072-x
  • Dentcheva D, Helbig S. On variational principles, level sets, well-posedness, and ε-solutions in vector optimization. J Optim Theory Appl. 1996;89:325–349. doi: 10.1007/BF02192533
  • Du WS. On some nonlinear problems induced by an abstract maximal element principle. J Math Anal Appl. 2008;347:391–399. doi: 10.1016/j.jmaa.2008.06.020
  • Finet C, Quarta L, Troestler C. Vector-valued variational principles. Nonlinear Anal. 2003;52:197–218. doi: 10.1016/S0362-546X(02)00103-7
  • Flores-Bazán F, Gutiérrez C, Novo V. A Brézis-Browder principle on partially ordered spaces and related ordering theorems. J Math Anal Appl. 2011;375:245–260. doi: 10.1016/j.jmaa.2010.09.014
  • Göpfert A, Tammer C, Zălinescu C. On the vectorial Ekeland's variational principle and minimal point theorems in product spaces. Nonlinear Anal. 2000;39:909–922. doi: 10.1016/S0362-546X(98)00255-7
  • Gutiérrez C, Jiménez B, Novo V. A set-valued Ekeland's variational principle in vector optimization. SIAM J Control Optim. 2008;47:883–903. doi: 10.1137/060672868
  • Ha TXD. Some variants of the Ekeland variational principle for a set-valued map. J Optim Theory Appl. 2005;124:187–206. doi: 10.1007/s10957-004-6472-y
  • Hamel AH. Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913–924. doi: 10.1016/j.na.2005.04.011
  • Isac G. The Ekeland's principle and the Pareto ε-efficiency. In: Tamiz M, editor. Multi-objective programming and goal programming: theories and applications. Berlin: Springer-Verlag; 1996. p. 148–163. (Lecture notes in econom. and math. systems; vol. 432).
  • Khanh PQ, Quy DN. Versions of Ekeland's variational principle involving set perturbations. J Glob Optim. 2013;57:951–968. doi: 10.1007/s10898-012-9983-3
  • Li SJ, Yang XQ, Chen GY. Vector Ekeland variational principle. In: Giannessi F, editor. Vector variational inequalities and vector equilibria. Dordrecht: Klower; 2000. p. 321–333. (Nonconvex optimization and its applications; vol. 38).
  • Li SJ, Zhang WY. A minimization theorem for a set-valued mapping. Appl Math Lett. 2008;21:769–773. doi: 10.1016/j.aml.2007.08.003
  • Liu CG, Ng KF. Ekeland's variational principle for set-valued functions. SIAM J Optim. 2011;21:41–56. doi: 10.1137/090760660
  • Németh AB. A nonconvex vector minimization problem. Nonlinear Anal. 1986;10:669–678. doi: 10.1016/0362-546X(86)90126-4
  • Qiu JH. A generalized Ekeland vector variational principle and its applications in optimization. Nonlinear Anal. 2009;71:4705–4717. doi: 10.1016/j.na.2009.03.034
  • Qiu JH. On Ha's version of set-valued Ekeland's variational principle. Acta Math Sin (Engl Ser). 2012;28:717–726. doi: 10.1007/s10114-011-0294-2
  • Qiu JH. Set-valued quasi-metrics and a general Ekeland's variational principle in vector optimization. SIAM J Control Optim. 2013;51:1350–1371. doi: 10.1137/110824115
  • Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937. doi: 10.1016/j.jmaa.2014.05.027
  • Qiu JH. An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems. Nonlinear Anal. 2016;27:26–42. doi: 10.1016/j.nonrwa.2015.07.005
  • Qiu JH. An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems. Acta Math Sin (Engl Ser). 2017;33:210–234. doi: 10.1007/s10114-016-6184-x
  • Qiu JH, He F. A general vectorial Ekeland's variational principle with a p-distance. Acta Math Sin (Engl Ser). 2013;29:1655–1678. doi: 10.1007/s10114-013-2284-z
  • Qiu JH, Li B, He F. Vectorial Ekeland's variational principle with a w-distance and its equivalent theorems. Acta Math Sci Ser B. 2012;32:2221–2236. doi: 10.1016/S0252-9602(12)60172-6
  • Tammer C. A generalization of Ekeland's variational principle. Optimization. 1992;25:129–141. doi: 10.1080/02331939208843815
  • Tammer C, Zălinescu C. Vector variational principle for set-valued functions. Optimization. 2011;60:839–857. doi: 10.1080/02331934.2010.522712
  • Zhu J, Zhong CK, Cho YJ. A generalized variational principle and vector optimization. J Optim Theory Appl. 2000;106:201–218. doi: 10.1023/A:1004619426652
  • Kuroiwa D. On set-valued optimization. Nonlinear Anal. 2001;47:1395–1400. doi: 10.1016/S0362-546X(01)00274-7
  • Gutiérrez C, Novo V, Ródenas-Pedregosa JL, et al. Nonconvex separation functional in linear spaces with applications to vector equilibria. SIAM J Optim. 2016;26:2677–2695. doi: 10.1137/16M1063575
  • Gerstewitz (Tammer) Chr. Nichtkonvexe Dualität in der Vektoroptimierung. Wiss Z TH Leuna-Merseburg. 1983;25:357–364.
  • Gerstewitz (Tammer) Chr, Iwanow E. DualitäT für nichtkonvexe Vektorptimierungsprobleme. Wiss Z Tech Hochsch Ilmenau. 1985;2:61–81.
  • Gerth C, Weidner P. Nonconvex separation theorems and some applications in vector optimization. J Optim Theory Appl. 1990;67(2):297–320. doi: 10.1007/BF00940478
  • Luc DT. Theory of vector optimization. Berlin: Springer; 1989. (Lecture notes in econom. and math. system; 319).
  • Saxon SA, Sánchez Ruiz LM. Dual local completeness. Proc Amer Math Soc. 1997;125:1063–1070. doi: 10.1090/S0002-9939-97-03864-1
  • Chen Y, Cho YJ, Yang L. Note on the results with lower semi-continuity. Bull Korean Math Soc. 2002;39:535–541. doi: 10.4134/BKMS.2002.39.4.535
  • Köthe G. Topological vector spaces I. Berlin: Springer-Verlag; 1969.
  • Kelley JL, Namioka I, Donoghue WF, Jr., et al. Linear topological spaces. Princeton: Van Nostrand; 1963.
  • Wilansky A. Modern methods in topological vector spaces. New York: McGraw-Hill; 1978.
  • Tammer C, Zălinescu C. Lipschitz properties of the scalarization function and applications. Optimization. 2010;59:305–319. doi: 10.1080/02331930801951033
  • Adán M, Novo V. Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. European J Oper Res. 2003;149:641–653. doi: 10.1016/S0377-2217(02)00444-7
  • Adán M, Novo V. Proper efficiency in vector optimization on real linear spaces. J Optim Theory Appl. 2004;121:515–540. doi: 10.1023/B:JOTA.0000037602.13941.ed
  • Qiu JH. The domination property for efficiency and Bishop-Phelps theorem in locally convex spaces. J Math Anal Appl. 2013;402:133–146. doi: 10.1016/j.jmaa.2012.12.072
  • Pérez Carreras P, Bonet J. Barrelled locally convex spaces. Amsterdam: North-Holland; 1987.
  • Holmes RB. Geometric functional analysis and its applications. New York: Springer; 1975.
  • Zălinescu C. Convex analysis in general vector spaces. Singapore: World Sci.; 2002.
  • Gutiérrez C, Jiménez B, Novo V. On approximate efficiency in multiobjective programming. Math Methods Oper Res. 2006;64:165–185. doi: 10.1007/s00186-006-0078-0
  • Gutiérrez C, Jiménez B, Novo V. A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J Optim. 2006;17:688–710. doi: 10.1137/05062648X

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