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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 73, 2024 - Issue 3
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Research Article

Equilibrium set-valued variational principles and the lower boundedness condition with application to psychology

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Pages 673-705 | Received 27 Dec 2021, Accepted 26 Aug 2022, Published online: 16 Sep 2022

References

  • Ekeland I. On the variational principle. J Math Anal Appl. 1974;47(2):324–353.
  • Chen GY, Huang XX, Yang XG. Vector optimization. Berlin: Springer-Verlag; 2005. (Set-Valued and Variational Analysis).
  • Ekeland I. Nonconvex minimization problems. Bull Amer Math Soc. 1979;1(3):443–474.
  • Göpfert A, Riahi H, Tammer C, et al. Variational methods in partially ordered spaces. New York: Springer-Verlag; 2003.
  • Bao TQ, Mordukhovich BS. Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math Program. 2010;122(2):301–347.
  • Bednarczuk EM, Zagrodny D. Vector variational principle. Arch Math (Basel). 2009;93(6):577–586.
  • Dentcheva D, Helbig S. On variational principles, level sets, well-posedness, and ϵ-solutions in vector optimization. J Optim Theory Appl. 1996;89(2):325–349.
  • Finet C, Quarta L, Troestler C. Vector-valued variational principles. Nonlinear Anal. 2003;52(1):197–218.
  • 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(1):245–260.
  • Göpfert A, Tammer C, Zalinescu C. On the vectorial Ekeland's variational principle and minimal point theorems in product spaces. Nonlinear Anal. 2000;39(7):909–922.
  • Gutiérrez C, Jiménez B, Novo V. A set-valued Ekeland's variational principle in vector optimization. SIAM J Control Optim. 2008;47(2):883–903.
  • Ha TXD. Some variants of the Ekeland variational principle for a set-valued map. J Optim Theory Appl. 2005;124(1):187–206.
  • Hamel AH. Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62(5):913–924.
  • Khanh PQ, Quy DN. Versions of Ekeland's variational principle involving set perturbations. J Glob Optim. 2013;57(3):951–968.
  • Liu CG, Ng KF. Ekeland's variational principle for set-valued functions. SIAM J Optim. 2011;21(1):41–56.
  • Németh AB. A nonconvex vector minimization problem. Nonlinear Anal. 1986;10(7):669–678.
  • Qiu JH. A generalized Ekeland vector variational principle and its applications in optimization. Nonlinear Anal. 2009;71(10):4705–4717.
  • Qiu JH. On Ha's version of set-valued Ekeland's variational principle. Acta Math Sinica English Ser. 2012;28(4):717–726.
  • Qiu JH. Set-valued quasi-metrics and a general Ekeland's variational principle in vector optimization. SIAM J Control Optim. 2013;51(2):1350–1371.
  • Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419(2):904–937.
  • Qiu JH. Generalized Gerstewitz functions and vector variational principle for ϵ-efficient solutions in the sense of Németh. Acta Math Sinica English Seri. 2019;35(3):297–320.
  • Qiu JH, He F. A general vectorial Ekeland's variational principle with a p-distance. Acta Math Sinica English Ser. 2013;29(9):1655–1678.
  • Tammer C. A generalization of Ekeland's variational principle. Optimization. 1992;25(2–3):129–141.
  • Tammer C, Zălinescu C. Vector variational principle for set-valued functions. Optimization. 2011;60(7):839–857.
  • Oettli W, Théra M. Equivalents of Ekeland's principle. Bull Austral Math Soc. 1993;48(3):385–392.
  • Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Student. 1994;63:123–145.
  • Al-Homidan S, Ansari QH, Yao JC. Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 2008;69(1):126–139.
  • Alleche B, RČŐdulescu VD. The Ekeland variational principle for equilibrium problems revisited and applications. Nonlinear Anal Real World Appl. 2015;23:17–25.
  • Bianchi M, Kassay G, Pini R. Existence of equilibria via Ekeland's principle. J Math Anal Appl. 2005;305(2):502–512.
  • Bianchi M, Kassay G, Pini R. Ekeland's principle for vector equilibrium problems. Nonlinear Anal. 2007;66(7):1454–1464.
  • Farkas C, Molnár AE. A generalized variational principle and its application to equilibrium problems. J Optim Theory Appl. 2013;156(2):213–231.
  • Gong X. Ekeland's principle for set-valued vector equilibrium problems. Acta Math Sci. 2014;34B:1179–1192.
  • Lin LJ, Du WS. Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J Math Anal Appl. 2006;323(1):360–370.
  • Qiu JH. An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems. Nonlinear Anal Real World Appl. 2016;27:26–42.
  • Qiu JH. An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems. Acta Math Sinica English Ser. 2017;33(2):210–234.
  • Zeng J, Li SJ. An Ekeland's variational principle for set-valued mappings with applications. J Comput Appl Math. 2009;230(2):477–484.
  • Bao TQ, Khanh PQ, Soubeyran A. Variational principles with generalized distances and the modelization of organizational change. Optimization. 2016;65(12):2049–2066.
  • Bao TQ, Mordukhovich BS, Soubeyran A. Variational analysis in psychological modeling. J Optim Theory Appl. 2015;164(1):290–315.
  • Bao TQ, Mordukhovich BS, Soubeyran A. Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality. Set-Valued Var Anal. 2015;23(2):375–398.
  • Bao TQ, Mordukhovich BS, Soubeyran A. Minimal points, variational principles, and variable preferences in set optimization. J Nonlinear Convex Anal. 2015;16:1511–1537.
  • Bao TQ, Théra MA. On extended versions of Dancs-Hegedüs-Medvegyev's fixed point theorem. Optimization. 2017;66(6):875–887.
  • Cobzas S. Completeness in quasi-metric spaces and Ekeland variational principle.Topol Appl. 2011;158(8):1073–1084.
  • Cobzas S. Functional analysis in asymmetric normed spaces. Basel: Birkhäuser/Springer Basel AG; 2013. (Frontiers in Mathematics).
  • Karapinar E, Romaguera S. On the weak form of Ekeland's variational principle in quasi-metric spaces.Topol Appl. 2015;184:54–60.
  • Qiu JH, He F, Soubeyran A. Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem. Optimization. 2018;67(1):25–53.
  • Qiu JH, Soubeyran A, He F. Equilibrium versions of set-valued variational principles and their applications to organizational behavior. Optimization. 2020;69(12):2657–2693.
  • Soubeyran A. Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors, GREQAM, Aix-Marseillle University, 2009. Preprint.
  • Soubeyran A. Variational rationality and the ‘unsatisfied man’: routines and course pursuit between aspirations, capabilities and beliefs, GREQAM, Aix-Marseillle University, 2010. Preprint.
  • Soubeyran A. Variational rationality 1.a Proximal dynamics and stationary traps: when is it worthwhile to move? GREQAM-AMSE, Aix-Marseillle University, 2018. Preprint.
  • Soubeyran A. Variational rationality 1.b Proximal dynamics and stationary traps: when is it worthwhile to move? GREQAM-AMSE, Aix-Marseillle University, 2018. Preprint.
  • Adan M, Novo V. Proper efficiency in vector optimization on real linear spaces. J Optim Theory Appl. 2004;121(3):515–540.
  • Qiu JH. The domination property for efficiency and Bishop-Phelps theorem in locally convex spaces. J Math Anal Appl. 2013;402(1):133–146.
  • Köthe G. Topological vector spaces I. Berlin: Springer-Verlag; 1969.
  • Kelley JL, Namioka I, Donoghue, Jr. WF, et al. Linear topological spaces. Princeton (NJ): Van Nostrand; 1963.
  • Gerstewitz (Tammer) C. Nichtkonvexe dualität in der vektoroptimierung. Wiss Z TH Leuna-Merseburg. 1983;25:357–364.
  • Fakhar M, Khodakhah MR, Soubeyran A, et al. Robust Ekeland variational principles. Application to the formation and stability of partnerships. Optimization, Published online: 3 Dec 2021.DOI: 10.1080/02331934.2021.2009626 null
  • Gutiérrez C, Jiménez B, Novo V. On approximate efficiency in multiobjective programming. Math Methods Oper Res. 2006;64(1):165–185.
  • Gutiérrez C, Jiménez B, Novo V. A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J Optim. 2006;17(3):688–710.
  • Soubeyran A. Variational rationality. Worthwhile stay and change approach-avoidance human dynamics ending in traps. GREQAM, AMSE, Aix Marseillle University. 2016. Preprint.
  • Soubeyran A. Variational rationality: towards a grand theory of motivation driven by worthwhile moves. AMSE, Aix-Marseille University. 2021. Preprint.
  • Soubeyran A. Variational rationality. The concepts of motivation and motivational force. AMSE. Aix-Marseille University. 2021. Preprint.
  • Soubeyran A. Variational rationality. The resolution of goal conflicts via stop and go approach-avoidance dynamics. AMSE, Aix-Marseille University. 2021. Preprint.
  • Soubeyran A. Variational rationality. A general theory of moving goals and intentions as satisficing worthwhile moves. AMSE, Aix-Marseille University. 2021. Preprint.
  • Soubeyran A. Variational rationality. Self regulation processes as satisficing variational principles and flexible/inexact algorithms. 2021. In preparation.
  • Lewin K. A dynamic theory of personality. New York (NY): McGraw Hill; 1935.
  • Lewin K. Principles of topological psychology. New York (NY): McGraw Hill; 1936.
  • Lewin K. The conceptual representation and measurement of psychological forces. Durham (NC): Duke University Press; 1938.
  • Lewin K. Field theory in social science. In Cartwright D, editor. Selected theoretical papers. New York (NY): Harper; 1951 & Row. Reprinted in Resolving Social Conflicts & Field Theory in Social Science. Washington (DC): American Psychological Association; 1997.
  • Simon HA. A behavioral model of rational choice.Q J Econ. 1955;69(1):99–118.
  • Fisher S, Cooper CL. On the move: the psychology of change and transition. New York (NY): Wiley; 1990.
  • Bao TQ, Mordukhovich BS, Soubeyran A, et al. Vector optimization with domination structures: variational principles and applications. Accepted Set Valued Anal Appl. 2022;30(2):695–729.
  • Samuelson W, Zeckhauser R. Status quo bias in decision making. J Risk Uncertainty. 1988;1(1):7–59.

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