Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type Variational Principle

Positivity

Volumn 9, Issue 3, 2005, Pages 511-539

  Isac, G ^a   Tammer, Chr ^b  

^a ROYAL MILITARY COLLEGE OF CANADA   (Canada)

^b MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27644557741     PISSN: 13851292     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11117-004-2770-8     Document Type: Article

References (58)

1. Attouch, H. and Riahi, H.: Stability results for Ekeland's ε-variational principle and cone extremal solutions, Math. Oper. Res. 18 (1993), 173-201.
2. Bahya, A. O.: Ensembles côniquement bornés et cônes nucléaires dans les espaces localement convexes séparés, These (Docteur de 3-éme cycle), Ecole Normale Supérieure, Takaddoum, Rabat, Maroc (Morocco), 1989.
3. Bahya, A. O.: Etude des cônes nucléaire, Ann. Sci. Math. Québec 15(2) (1992), 123-134.
4. Bahya, A. O.: Etude des cônes nucléaires dans les espaces localement convexes séparés, Thèse de Docteur d'Etat, Faculté des Sciences de Rabat, Maroc (Morocco), 1997.
5. Blum, E. and Oettli, W.: Mathematische Optimierung, Springer-Verlag, Heidelberg, 1975.
6. Borwein, J.: Continuity and differentiability properties of convex operators, Proc. London Math. Soc. 44 (1982), 420-444.
7. Borwein, J.: On the existence of Pareto efficient points, Math. Oper. Res. 8 (1983), 68-75.
8. Cammaroto, F. and Chinni, A.: A complement to Ekeland's variational principle in Banach spaces, Bull. Polish Acad. Sci. Math. 44 (1) (1996), 29-33.
9. Cammaroto, F., Chinni, A and Sturiale, G.: A complement to Ekeland's variational principle for vector-valued functions, (Preprint, University of Messina).
10. Cammaroto, F., Chinni, A and Sturiale, G.: A remark on Ekeland's principle in locally convex topological vector spaces, (Preprint, University of Messina).
11. Chen, G.Y. and Cheng, G.M.: vector variational inequality and vector optimization. In Sawaragi, Y., Inoue,K and Nakayama, H (eds.) Interactive and Intelligent Decision Support Systems, Vol. I, Proceedings 1986. Lecture Notes in Econ. and Math. Syst. 285, Springer-Verlag, Berlin Heidelberg, New York, 1986.
12. Chen, G.Y. and Craven B.D.: A vector variational inequality and optimization over an efficient set, ZOR-Meth. Model Oper. Res. 34 (1990), 1-12.
13. Chen, G.Y. and Huang, X.X.: A unified approach to the existing three types of variational principles for vector valued functions, Math. Meth. Oper. Res. 48 (1998), 349-357.
14. Chen, G.Y, Huang, X.X. and Han, S.H.: General Ekeland's variational principle for set-valued mappings, J. Opt. Theory Appl. 106 (1) (2000), 151-164.
15. De Figueiredo, D. G.: The Ekeland's variational principle with applications and detours, Tata Institute of Fundamental Research, Bombay, (1989).
16. Ekeland, I.: Sur les problemes variationnels, C. R. Acad. Sci. Paris 275 (1972), A1057-1059.
17. Ekeland, I.: On some variational principle, J. Math. Anal. Appl. 47 (1974), 324-354.
18. Ekeland, I.: Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (3), (1979), 443-474.
19. Ekeland, I.: Some lemmas about dynamical systems, Math. Scand. 52 (1983), 262-268.
20. Ekeland, I.: The ε-variational principle revised, (Notes by S. Terracini), Methods of Noncovex Analysis A. Cellina (ed.) Lecture Notes in Math. Springer-Verlag, 1446 (1990), pp. 1-15.
21. Fang, J.X.: The variational principle and fixed point theorems in certain topological spaces, J. Math. Anal. Appl. 208 (1996), 389-412.
22. Georgiev, P. G.: The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal Appl. 131 (1988), 1-21.
23. Gerth (Tammer), Chr., and Weidner, P.: Nonconvex separation theorems and some applications in vector optimization J. Opt. Theory Appl. 67 (2) (1990), 297-320.
24. Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems, in: R.W. Cottle, F. Giannessi, J.L. Lions, (eds) Variational Inequality and Complementarity Problems, J. Wiley and Sons, 1980 pp. 151-186.
25. Göpfert, A. and Tammer, Chr.: A new maximal point theorem, Z. Anal. Anwendungen. 14 (1995), 379-390.
26. Göpfert, A. and Tammer, Chr.: Maximal point theorems in product spaces and applications for multicriteria approximation problems, In: Y.Y. Haimes, R.E. Stener (eds) Research and Practice in Multiple Criteria Decision Making. Lecture Notes in Econ. and Math. Systems, 487, Springer, Berlin Heidelberg, 1999, pp. 93-104.
27. Göpfert, A., Tammer, Chr. and Žlinescu, C.: On the vectorial Ekeland's variational principle and minimal points in product spaces, Nonlinear Anal, 39 (2000), 909-922.
28. Göpfert, A., Tammer, Chr. and Zǎlinescu, C.: A new maximal point theorem in product spaces, ZAA 18(3), (1999), 767-770.
29. Hyers, D. H. Isac, G. and Rassias, Th. M.: Topics in nonlinear analysis and applications, World Scientific (1997).
30. Isac, G.: Cônes localement borné et cônes completement reguliers. Applications à l'Analyse Nonlinéaire, Séminaire D'Analyse Moderne, Université de Sherbrooke, 17 (1980).
31. Isac, G.: Sur l'existence de l'optimum de Pareto, Riv. Mat. Univ. Parma (4) 9 (1983), 303-325.
32. Isac, G.: Un critère de sommabilité dans les espaces localement convexes ordonnes: Cônes nucléaires, Mathematica 25 (48), 2 (1983), 159-169.
33. Isac, G.: Supernormal cone and fixed point theory, Rocky Mountain J. Math. 17 (3) (1987), 219-226.
34. Isac, G.: The Ekeland's principle and the Pareto ε-efficiency, In: M. Tamiz (ed) Multi-objective Programming and Goal Programming. Theory and Applications, Lecture Notes in Economics and Mathematical Systems, Vol. 432, Springer-Verlag, 1996, 148-163.
35. Isac, G.: Ekeland's principle and nuclear cones: a geometrical aspect, Math. Comput. Modeling, 26 (11) (1997), 111-116.
36. Isac, G.: On Pareto efficiency. A general constructive existence principle, To appear in the volume: Combinatorial and Global Optimization (eds), P.M. Pardalos, A. Migdalas and R. Burkard, Kluwer Academic Publishers, 133-144.
37. Isac, G. and Bahya, A. O.: Full nuclear cones associated to a normal cone. Application to Pareto efficiency, Appl Math. Lett. 15 (2002), 633-639.
38. Jahn, J.: Mathematical Vector Optimization in Partially Ordered Spaces, Lang Verlag Frankfurt, Bern, New York, 1986.
39. 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, 1986.
40. Mancino, O. G. and Stampacchia, G.: Convex programming and variational inequalities, J. Opt. Theory Appl. 9 (1) (1972), 3-23.
41. Nemeth, A.: A nonconvex vector minimization problem, Nonlinear Anal. Theory, Meth. Appl. 10 (1986), 669-678.
42. Nemeth, A.: Ekeland's variational principle in ordered abelian groups, forthcoming: Nonlinear Anal. Forum.
43. Oettli, W.: Approximate solutions of variational inequalities. In: H. Albach, E. Helmstädter, and R. Henn (eds) Quantitative Wirtschaftsforschung, Verlag J. C. B. Mohr, Tübingen 1977, 535-538.
44. Oettli, W. and Thera, M.: Equivalents of Ekeland's principle, Bull. Austral. Math. Soc. 48 (1993), 385-392.
45. Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability (2nd ed.), Lecture Notes Math. 1364, Springer-Verlag, 1993.
46. Pontini, C.: Inclusion theorems for non-explosive and strongly exposed cones in normed spaces, J. Math. Anal. Appl. 148 (1990), 275-286.
47. Tammer, Chr.: Existence results and necessary conditions for ε-efficient elements, In: B. Brosowski, J. Ester, S. Helbig and R. Nehse Multicriteria Decision Frankfurt am Main: Verlag P. Lang, 1993, pp. 97-109.
48. Tammer, Chr.: A generalization of Ekeland's variational principle, Optimization 25 (1992), 129-141.
49. Treves, F.: Locally Convex Spaces and Linear Partial Differential Equations, Springer-Verlag, New-York 1967.
50. Truong, X.D.H.: On the existence of efficient points in locally convex spaces, J. Global Opt. 4 (1994), 265-278.
51. Truong, X.D.H.: A note on a class of cones ensuring the existence of efficient points in bounded complete sets, Optimization 31 (1994), 141-152.
52. Truong, X. D. H.: Existence and density results for proper efficiency in cone compact sets (Preprint, Hanoi Institute of Mathematics), 1999.
53. Turinici, M.: Vector extensions of the variational Ekeland's result, Ann. St. Univ. Al. I. Cuza, Iasi, Tom XL, Matematica, F-3 (1994), 225-266.
54. Yang, X.Y.: Vector variational inequality and its duality. Nonlinear Anal. Theory Meth. Appl. 21(11) (1993), 869-877.
55. Yang X.Q. and Chen G.Y: On inverse vector variational inequalities, In F. Giannessi, (ed.) Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht, (2000), 433-446.
56. Yang X.Q. and Goh, C. J.: On vector variational inequalities: application to vector equilibria. J. Opt. Theory Appl. 95 (1997), 431-443.
57. Yang X.Q. and Goh, C.J.: Vector Variational Inequalities, Vector Equilibrium Flow and Vector Optimization, in F. Giannessi, (ed.) Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, Nonconvex Optimization and its Applications, Kluwer Academic Publishers. Dordrecht, 2000, 447-465.
58. Zhu, J., Isac, G. and Zhao, D.: Pareto optimization in topological vector spaces, (Preprint, 2001).