Abstract
We present in this paper a new variant of Ekeland’s variational principle for vector valued functions with applications to the study of Pareto ε-efficiency. A new existence result for Pareto efficiency is also presented.
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References
Altman M.: A generalization of Brézis-Browder principle on ordered sets, Nonlin. Anal., Theory, Meth. and Appl. 6 Nr. 2 (1982), 157–165.
Banas J.: On drop property and nearly uniformly smooth Banach spaces, Nonlin. Anal., Theory, Meth. and Appl. 14 (1990), 927–933.
Brézis H. and Browder F. E.: A general principle on ordered set in nonlinear functional analysis, Adv. Math. 21 (1976), 777–787.
Caristi J.: Fixed point theorems for mappings satisfying inwardness condition, Trans. Amer. Math. Soc, 215(1976),241–251.
Danes J.: A geometric theorem useful in nonlinear functional analysis, Boll. Un. Mat. Ital. 6 (1972), 369–375.
Danes J.: Equivalence of some geometric and related results of nonlinear functional analysis, Comm. Math. Univ. Carolinae 26(1985), 445–454.
Dancs S., Hegedus M. and Medvegyev P.: A general ordering and fixed-point principle in complete metric space, Acta Sci. Math. (Szeged) 46(1983), 381–388.
De Figueiredo D. G.: The Ekeland variational principle with applications and detours, Tata Institute of Fundamental Research, Bombay (1989)
Ekeland L: Sur les problèmes variationnels, C.R. Acad. Sci. Paris 275 (1972), A 1057–1059.
Ekeland L: On some variational principle, J. Math. Anal. Appl. 47 (1974), 324–354.
Ekeland L: Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (3), (1979), 443–474.
Ekeland L: Some lemmas about dynamical systems, Math. Scand. 52 (1983), 262–268.
Ekeland L: The s-variational principle revised, (Notes by S. Terracini), Methods of nonconvex analysis (Ed. A. Cellina), Lecture Notes in Math. Springer-Verlag Nr. 1446 (1990), 1–15.
Elliot R. J. and Jarvis T. M.: Prior play in a deterministic differential gaine, J. Math. Anal. Appl. 86 (1982), 137–145.
Georgiev P. G.: The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988), 1–21.
Giles J. R. and Kutzarova D. N.: Characterization of drop and weak drop properties for closes bounded convex sets, Bull. Austral. Math. Soc. 43 (1991), 337–385.
Giles J. R., Sims B. and Yorke A. C.: On the drop and weak drop properties for a Banach space, Bull. Austral. Math. Soc. 41 (1990), 503–507.
Isac G.: Sur l’existence de l’optimum de Pareto, Riv. Mat. Univ. Parma (4) 9 (1983), 303–335.
Isac G.: Pareto optimization in ifnfinite dimensional spaces: the importance of nuclear cones, J. Math. Anal. Appl. 182(1994), 393–404.
John J.: Mathematical vector optimization in partially ordered linear spaces, Peter Lang, Frankfurt (1986).
Khanh P. Q.: On Caristi-Kirk’s theorem and Ekeland’s variational principle for Pareto extrema, Preprint 357 Institute of Mathematics, Polish Academy of Sciences.
Loridan P.: ε-solutions in vector minimization problems, J. Optim. Theory Appl. 43 Nr. 2 (1984), 265–276.
Luc D. T.: Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag Nr. 319 (1989).
Montesinos V.: Drop property equals reflexivity, Studia Math. 87 (1987), 93–100.
Németh A. B.: A Nonconvex vector minimization problem, Nonlin. Anal. Theory, Methods and Appl. 10 (1986), 669–678.
Németh A. B.: Between Pareto efficiency and Pareto s-efficiency, Optimization 20 Nr. 5 (1989), 615–637.
Oettli W. and Théra M.: Equivalents of Ekeland’s principle, Bull. Austral. Math. Soc. 48 (1993), 385–392.
Penot J. P.: The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlin. Anal., Theory, Meth., Appl. 10 Nr. 9 (1986), 813–822.
Perressini A. L.: Ordered topological vector spaces, Harper and Row (1967), New York, Evanston and London.
Rolewicz S.: On drop property, Studia Math. 85 (1987), 27–35.
Rolewicz S.: On A-uniform convexity and drop property, Studia Math, 87 (1987), 181–191.
Schaefer H. H.: Topological vector spaces Mcmillan Company, New York, London (1966).
Staib T.: On two generalizations of Pareto minimality, J. Optim. Theory Appl. 59 Nr. 2 (1988), 289–306.
Takahashi W.: Existence theorems generalizing fixed point theorems for multivalued mappings, Fixed point theory and applications (ed. J. B. Baillon and M. Théra) Pitman Research Notes in Math., 252, Longman, Harlow (1991), 397–406.
Tammer Chr.: A generalization of Ekeland’s variational principle, Optimization, 25 (1992), 129–141.
Tammer Chr.: A variational principle and a fixed point theorem, To appear (Proc. IFIP-Conf Compiegne (1993)).
Tammer Chr.: existence results and necessary conditions for ε-efficient elements, In: B. Brosowski, J. Ester, S. Helding and R. Nehse (eds), Multicriteria Decision. Proc. 14 Th. Meeting of the German Working Group “ Mehrkriterielle Entsheidung” Peter Larg, Frankfurt (1993), 97–110.
Valyi L: Approximate saddle-point theorems in vector optimization, J. Optim. Theory Appl. 55 Nr. 3 (1987), 435–448.
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Isac, G. (1996). The Ekeland’s Principle and the Pareto ε-Efficiency. In: Tamiz, M. (eds) Multi-Objective Programming and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol 432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87561-8_12
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DOI: https://doi.org/10.1007/978-3-642-87561-8_12
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