References
H. Attouch H. Riahi (1993) ArticleTitleStability results for Ekeland’s ε-variational principle and cone extremal solutions Math. Oper. Res. 18 173–201
Bahya A.O. Ensembles côniquement bornés et cônes nucléaires dans les espaces localement convexes séparés. These (Docteur de 3-ème cycle), Ecole Normale Supérieure, Takaddoum, Rabat, Maroc (Morocco), 1989.
A.O. Bahya (1992) ArticleTitleEtude des cônes nucléaire Ann. Sci. Math. Qué bec 15 IssueID2 123–134
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.
E. Blum W. Oettli (1975) Mathematische Optimierung Springer-Verlag Heidelberg
J. Borwein (1982) ArticleTitleContinuity and differentiability properties of convex operators Proc. London Math. Soc. 44 420–444
J. Borwein (1983) ArticleTitleOn the existence of Pareto efficient points Math. Oper. Res. 8 68–75
F. Cammaroto A. Chinni (1996) ArticleTitleA complement to Ekeland’s variational principle in Banach spaces Bull. Polish Acad. Sci. Math. 44 IssueID1 29–33
Cammaroto F., Chinni A., Sturiale G.: A complement to Ekeland’s variational principle for vector-valued functions, (Preprint, University of Messina).
Cammaroto F., Chinni A. and Sturiale G.: A remark on Ekeland’s principle in locally convex topological vector spaces, (Preprint, University of Messina).
G.Y. Chen G.M. Cheng (1986) vector variational inequality and vector optimization Y. Sawaragi K. Inoue H Nakayama (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
G.Y. Chen B.D. Craven (1990) ArticleTitleA vector variational inequality and optimization over an efficient set ZOR-Meth. Model Oper. Res. 34 1–12 Occurrence Handle10.1007/BF01415945
G.Y. Chen X.X. Huang (1998) ArticleTitleA unified approach to the existing three types of variational principles for vector valued functions Math. Meth. Oper. Res. 48 349–357 Occurrence Handle10.1007/s001860050032
G.Y. Chen X.X. Huang S.H. Han (2000) ArticleTitleGeneral Ekeland’s variational principle for set-valued mappings J. Opt. Theory Appl. 106 IssueID1 151–164 Occurrence Handle10.1023/A:1004663208905
D.G. Figueiredo ParticleDe (1989) The Ekeland’s variational principle with applications and detours Tata Institute of Fundamental Research Bombay
I. Ekeland (1972) ArticleTitleSur les problemes variationnels C. R. Acad. Sci. Paris 275 A1057–1059
I. Ekeland (1974) ArticleTitleOn some variational principle J. Math. Anal. Appl. 47 324–354 Occurrence Handle10.1016/0022-247X(74)90025-0
I. Ekeland (1979) ArticleTitleNonconvex minimization problems Bull. Amer. Math. Soc. 1 IssueID3 443–474
I. Ekeland (1983) ArticleTitleSome lemmas about dynamical systems Math. Scand. 52 262–268
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.
J.X. Fang (1996) ArticleTitleThe variational principle and fixed point theorems in certain topological spaces J. Math. Anal. Appl. 208 389–412
P.G. Georgiev (1988) ArticleTitleThe strong Ekeland variational principle, the strong drop theorem and applications J. Math. Anal. Appl. 131 1–21 Occurrence Handle10.1016/0022-247X(88)90187-4
Chr. Gerth (Tammer) P. Weidner (1990) ArticleTitleNonconvex separation theorems and some applications in vector optimization J. Opt. Theory Appl. 67 IssueID2 297–320 Occurrence Handle10.1007/BF00940478
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.
A. Göpfert Chr. Tammer (1995) ArticleTitleA new maximal point theorem, Z Anal. Anwendungen. 14 379–390
A. Göpfert Chr. Tammer (1999) Maximal point theorems in product spaces and applications for multicriteria approximation problems 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 93–104
A. Göpfert Chr. Tammer C. Zălinescu (2000) ArticleTitleOn the vectorial Ekeland’s variational principle and minimal points in product spaces Nonlinear Anal 39 909–922 Occurrence Handle10.1016/S0362-546X(98)00255-7
A. Göpfert Chr. Tammer C. Zălinescu (1999) ArticleTitleA new maximal point theorem in product spaces ZAA 18 IssueID3 767–770
Hyers, D. H. Isac, G. and Rassias, Th. M.: Topics in nonlinear analysis and applications, World Scientific (1997).
Isac, G.: Cônes localement borné et cônes completement reguliers. Applications à l’Analyse Nonlinéaire, Séminaire D’Analyse Moderne, Université de Sherbrooke, 17 (1980).
G. Isac (1983) ArticleTitleSur l’existence de l’optimum de Pareto Riv. Mat. Univ. Parma 4 IssueID9 303–325
G. Isac (1983) ArticleTitleUn critère de sommabilité dans les espaces localement convexes ordonnes: Cônes nucléaires Mathematica 25(48) 2 159–169
G. Isac (1987) ArticleTitleSupernormal cone and fixed point theory Rocky Mountain J. Math. 17 IssueID3 219–226
Isac, G.: The Ekeland’s principle and the Pareto ε-efficiency, In: M. Tamiz (ed) Multiobjective Programming and Goal Programming. Theory and Applications, Lecture Notes in Economics and Mathematical Systems, Vol. 432, Springer-Verlag, 1996, 148–163.
G. Isac (1997) ArticleTitleEkeland’s principle and nuclear cones: a geometrical aspect Math. Comput. Modeling 26 IssueID11 111–116 Occurrence Handle10.1016/S0895-7177(97)00223-9
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.
G. Isac A.O. Bahya (2002) ArticleTitleFull nuclear cones associated to a normal cone. Application to Pareto efficiency Appl Math. Lett. 15 633–639 Occurrence Handle10.1016/S0893-9659(02)80017-9
J. Jahn (1986) Mathematical Vector Optimization in Partially Ordered Spaces Lang Verlag Frankfurt Bern, New York
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.
O.G. Mancino G. Stampacchia (1972) ArticleTitleConvex programming and variational inequalities J. Opt. Theory Appl. 9 IssueID1 3–23 Occurrence Handle10.1007/BF00932801
A. Nemeth (1986) ArticleTitleA nonconvex vector minimization problem Nonlinear Anal. Theory, Meth. Appl. 10 669–678
Nemeth, A.: Ekeland’s variational principle in ordered abelian groups, forthcoming: Nonlinear Anal. Forum.
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.
W. Oettli M. Thera (1993) ArticleTitleEquivalents of Ekeland’s principle Bull. Austral. Math. Soc. 48 385–392
Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability (2nd ed.), Lecture Notes Math. 1364, Springer-Verlag, 1993.
C. Pontini (1990) ArticleTitleInclusion theorems for non-explosive and strongly exposed cones in normed spaces J. Math. Anal. Appl 148 275–286 Occurrence Handle10.1016/0022-247X(90)90001-V
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.
Chr. Tammer (1992) ArticleTitleA generalization of Ekeland’s variational principle Optimization 25 129–141
F. Treves (1967) Locally Convex Spaces and Linear Partial Differential Equations Springer-Verlag New-York
X.D.H. Truong (1994) ArticleTitleOn the existence of efficient points in locally convex spaces J. Global Opt. 4 265–278 Occurrence Handle10.1007/BF01098361
X.D.H. Truong (1994) ArticleTitleA note on a class of cones ensuring the existence of efficient points in bounded complete sets Optimization 31 141–152
Truong, X. D. H.: Existence and density results for proper efficiency in cone compact sets (Preprint, Hanoi Institute of Mathematics), 1999.
M. Turinici (1994) ArticleTitleVector extensions of the variational Ekeland’s result Ann. St. Univ. Al. I. Cuza, Iasi, Tom XL, Matematica F-3 225–266
X.Y. Yang (1993) ArticleTitleVector variational inequality and its duality Nonlinear Anal. Theory Meth. Appl. 21 IssueID11 869–877 Occurrence Handle10.1016/0362-546X(93)90052-T
X.Q. Yang G.Y. Chen (2000) On inverse vector variational inequalities F. Giannessi (Eds) Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Nonconvex Optimization and its Applications Kluwer Academic Publishers Dordrecht 433–446
X.Q. Yang C.J. Goh (1997) ArticleTitleOn vector variational inequalities: application to vector equilibria J. Opt. Theory Appl. 95 431–443 Occurrence Handle10.1023/A:1022647607947
X.Q. Yang C.J. Goh (2000) Vector Variational Inequalities, Vector Equilibrium Flow and Vector Optimization F. Giannessi (Eds) Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, Nonconvex Optimization and its Applications Kluwer Academic Publishers Dordrecht 447–465
Zhu, J., Isac, G. and Zhao, D.: Pareto optimization in topological vector spaces, (Preprint, 2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Isac, G., Tammer, C. Nuclear and Full Nuclear Cones in Product Spaces: Pareto Efficiency and an Ekeland Type Variational Principle. Positivity 9, 511–539 (2005). https://doi.org/10.1007/s11117-004-2770-8
Issue Date:
DOI: https://doi.org/10.1007/s11117-004-2770-8