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Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 63))

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Abstract

A very popular domain of applied mathematics is optimization, because the diversity of its applications to economics, engineering and sciences. Certainly the applications to practical problems stimulated the impressive development of this domain. Between the chapters of optimization, one is the optimization of vector-valued functions, known also under the name of Pareto optimization. In 1906 V. Pareto wrote: “Principeremo con deftnire un termine di cui è comodo fare uso per scansare lungaggini. Diremo che i componenti di una colletivita godono, in una certa postione, del massimo di ofelimita, quando è impossibile allontanarsi pochissimo da quella positione giovando, o nuocendo, a tutti i componenti la collectività; ogni picolissimo spostamento da quella positione avendo necessariamente per effetto di giovare a parte dei componenti la collectività e di nuocere ad altri.” (Pareto, 1919).

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References

  • Arrow, K.J., Barankin, E.W. and Blackwell, D. ‘Admissible Points of Convex Sets’.- In: Contributions to the Theory of Games, 2. K.W. Kuhn and A.W. Tucker, eds., Princeton University Press: Princeton, New Jersey, 1953.- P. 87–92

    Google Scholar 

  • Bacopoulos A, Godini G. and Singer I. Infima of sets in the plane and applications to vectorial optimization. Rev. Roumaine de Math. Pures et Appl. 1978; 23: 343–360

    MathSciNet  MATH  Google Scholar 

  • Bahya AO. 1. Étude des cônes nucléaires. Ann. Sci. Math. Quebec. 1991; 15: 123–133

    MathSciNet  MATH  Google Scholar 

  • Bahya AO. 2. Ensembles côniquement bornés et cônes nucléaires dans les éspaces localement convexes séparés’. Thèse (3-eme cycle), Ecole Normale Superieure, Rabat, Morocco, 1989.

    Google Scholar 

  • Bahya AO. 3. Résultats sur des cônes séquentiellement complètement réguliers’.Preprint, 1995.

    Google Scholar 

  • Bahya AO. 4. Études des Cônes Nucléaires dans les Éspaces Localement Convexes Séparés. Thèse (Docteur d’Etat Es. Sciences Mathématiques), Université Mohammed V, Faculté des Sciences, Rabat, Morocco, 1997.

    Google Scholar 

  • Becker R. Sur les cônes (faiblement complets) contenus dans le dual d’un espace de Banach non reflexif. Bull. Austral. Math. Soc. 1989; 39: 321–328.

    Article  MathSciNet  MATH  Google Scholar 

  • Benson HP. 1. An improved definition of proper efficiency for vector minimization with respect to cones. J. Math. Anal. Appl. 1979; 71: 232–241.

    Article  MathSciNet  MATH  Google Scholar 

  • Benson HP. 2. On a domination property for vector maximization with respect to cones. J. Optim. Theory Appl. 1983; 39: 125–132;

    Article  MathSciNet  MATH  Google Scholar 

  • Benson HP. 2. Errata Corrige, J. Optim. Theory Appl. 1984; 43: 477–479.

    Article  MathSciNet  Google Scholar 

  • Benson HP. 3. Efficiency and proper efficiency for vector minimization with respect to cones. J. Math. Anal. Appl. 1983; 93: 273–289.

    Article  MathSciNet  MATH  Google Scholar 

  • Benson HP and Morin TL. The vector maximization problem: proper efficiency and stability. SIAM J. Appl. Math. 1977; 32: 64–72.

    Article  MathSciNet  MATH  Google Scholar 

  • Bergstresser K, Charnes A and Yu PL. Generalization of domination structures and nondomi-nated solutions in multicriteria decision making. J. Optim. Theory Appl. 1976; 18: 3–13.

    Article  MathSciNet  MATH  Google Scholar 

  • Bitran GR and Magnanti TL. The structure of admissible points with respect to cone dominance. J. Optim. Theory Appl. 1979; 29: 573–614.

    Article  MathSciNet  MATH  Google Scholar 

  • Borwein JM. 1. Proper efficient points for maximization with respect to cones. SIAM J. Control Opt. 1977; 15: 57–63.

    Article  MATH  Google Scholar 

  • Borwein JM. 2. The geometry of Pareto efficiency over cones. Math. Operationsforsch. Scr. Optim. 1980; 11: 235–248.

    MATH  Google Scholar 

  • Borwein JM. 3. On the existence of Pareto efficient points. Math. Oper. Res. 1983; 8: 68–75.

    Google Scholar 

  • Borwein JM and Zhuang DM. 1. Super-efficiency in vector optimization. ZOR — Methods and Models of Operations Research. 1991; 35: 175–184.

    MathSciNet  MATH  Google Scholar 

  • Borwein JM and Zhuang DM. 2. Super-efficiency in vector optimization. Trans. Amer. Soc. 1993; 338: 105–122.

    Article  MathSciNet  MATH  Google Scholar 

  • Bucur I and Postolica V. A coincidence result between sets of efficient points and Choquet boundaries in separated locally convex spaces. Optimization. 1996; 36: 231–234.

    Article  MathSciNet  MATH  Google Scholar 

  • Cesary L and Suryanarayana MB. Existence theorems for Pareto optimization, multivalued and Banach space valued functional. Trans. Amer. Math. Soc. 1978; 244: 37–65.

    Article  MathSciNet  Google Scholar 

  • Chen Guang-Ya. On generalized Arrow-Barankin-Blackwell theorems in locally convex spaces. J. Optim. Theory Appl. (in press)

    Google Scholar 

  • Chen GY and Huang XX. 1. Stability results for Ekeland’s -varational principle for vector valued functions. Math. Meth. Oper. Res. 1998; 48: 97–103.

    Article  MATH  Google Scholar 

  • Chen GY and Huang XX. 2. A unified approach to the existing three types of variational principles for vector valued functions. Math. Meth. Oper. Res. 1998; 48: 349–357.

    Article  MATH  Google Scholar 

  • Chew KL. Maximal points with respect to cone dominance in Banach spaces and their existence. J. Optim. Theory Appl. 1984; 44: 1–53.

    Article  MathSciNet  MATH  Google Scholar 

  • Chichilnisky G and Kalman PJ. Applications of functional analysis to models of efficient allocation of economic resourses. J. Optim. Theory Appl. 1980; 30: 19–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Choo EV and Atkins DR. Connectedness in multiple linear fractional programming. Management Science. 1983; 29: 250–255.

    Article  MathSciNet  MATH  Google Scholar 

  • Choquet, G. Lectures on Analysis, Vol. 1–3, Mathematics Lecture Notes Series, New York-Amsterdam: Benjamin, 1969.

    Google Scholar 

  • Constantinescu G. L’ensemble des fonctions surharmoniques positives sur un esipace harmonique est faiblement complet. C.R.Acad. Sci. Paris. 1970; 271: A549-A551.

    MathSciNet  Google Scholar 

  • Corley HW. 1. An existence result for maximization with respect to cones. J. Optim. Theory Appl. 1980; 31: 277–281.

    Article  MathSciNet  MATH  Google Scholar 

  • Corley HW. 2. Some hybrid fixed point theorems related to optimization. J. Math. Anal. Appl. 1986; 120: 528–532.

    Article  MathSciNet  MATH  Google Scholar 

  • Dauer JP and Gallagher RJ. Positive proper efficient points and related cone results in vector optimization theory. SIAM J. Control Opt. 1990; 28: 158–172.

    Article  MathSciNet  MATH  Google Scholar 

  • Dauer JP and Saleh OA. A characterization of proper minimal points as solutions of sublinear optimization problems. J. Math. Anal. Appl. 1993; 178: 227–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Dauer JP and Stadler W. A survey of vector optimization in infinite dimensional spaces. Part II. J. Optim. Theory Appl. 1986; 51: 205–241.

    Article  MathSciNet  MATH  Google Scholar 

  • Durier R and Michelot C. Sets of efficient points in a normed space. J. Mith. Anal. Appl. 1986; 117: 506–528.

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards, R.E. Functional Analysis. New York: Holt Reinhart and Winston, 1965.

    MATH  Google Scholar 

  • Ekeland I. 1. Sur les problemes variationnels. C.R. Acad. Sci. Paris. 1972; 275: A1057-A1059.

    MathSciNet  Google Scholar 

  • Ekeland I. 2. On some variational principle. J. Math. Anal. Appl. 1974; 47: 324–354.

    Article  MathSciNet  MATH  Google Scholar 

  • Ekeland I. 3. Nonconvex minimization problems. Bull. Amer. Math. Soc. 1979; 1(3): 443–474.

    Article  MathSciNet  MATH  Google Scholar 

  • Ekeland I. 4. Some lemmas about dynamical systems. Math. Scan. 1983; 52: 262–268.

    MathSciNet  MATH  Google Scholar 

  • Ekeland I. 5. ‘The -variational principle revised’ (notes by S. Terracini). — In: Methods on Nonconvex Analysis, A. Cellina, ed., Lecture Notes in Math. Berlin-Heidelberg: Springer-Verlag, 1990, Nr.1446.- P. 1–15.

    Chapter  Google Scholar 

  • Fan K. Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 1953; 39: 42–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrot F. 1. A generalization of the Arrow-Barankin-Blackwell theorem in normed spaces. J. Math. Anal. Appl. 1991; 158: 47–54.

    Article  MathSciNet  Google Scholar 

  • Ferrot F. 2. General form of the Arrow-Barankin-Blackwell theorem in normed spaces and in the l -case. J. Math. Anal. Appl. 1993; 79: 127–138.

    Google Scholar 

  • Ferrot F. 3. A new ABB theorem in Banach spaces. Preprint, Universita di Genova, Italy, 1999.

    Google Scholar 

  • Fu WT. On the density of proper efficient points. Proceed. Amer. Math. Soc. 1996; 124: 1213–1217.

    Article  MATH  Google Scholar 

  • Gallagher RJ and Saleh OA. Two generalizations of a theorem of Arrow-Barankin-Blackwell. SIAM J. Control Opt. 1993; 31: 247–256.

    Article  MathSciNet  MATH  Google Scholar 

  • Georffrion AM. Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 1968; 22: 618–630.

    Article  MathSciNet  Google Scholar 

  • Gopfert A and Tammer Chr. 1. A new maximal point theorem. ZAA. 1995; 14: 379–390.

    MathSciNet  Google Scholar 

  • Gopfert A and Tammer Chr. 2. “-Approximate solutions and conicall support points. A new maximal point theorem”. Preprint, University of Halle, Germany, 1997.

    Google Scholar 

  • Gopfert A and Tammer Chr. 3. “Maximal point theorems in product spaces and applications for multicriteria approximation problems”. Preprint Nr.26, University of Halle, Germany, 1998.

    Google Scholar 

  • Gopfert A, Tammer Chr. and Zalinescu C. 1. “On the vectorial Ekeland’s variational principle and minimal points in product spaces.” Preprint, University of Halle, Germany, 1998.

    Google Scholar 

  • Gopfert A, Tammer Chr. and Zalinescu C. 2. “Maximal point theorems in product spaces.” Preprint, University of Halle, Germany, 1998.

    Google Scholar 

  • Gopfert A, Sekatzek M and Tammer Chr. “Novelties around Ekeland’s variational principle.” Preprint, University of Halle, Germany, 1999.

    Google Scholar 

  • Gong XH. Connectedness of efficient solution set for set-valued maps in normed spaces. J. Optim. Theory Appl. 1994; 83: 83–96.

    Article  MathSciNet  MATH  Google Scholar 

  • Gotz A and Jahn J. The Lagrange multiplier rule in set-valued optimization. (To appear in SIAM J. Opt.)

    Google Scholar 

  • Guerraggio A, Molho E and Zaffaroni A. On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 1994; 82: 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  • Hartley R. On cone-efficiency, cone-convexity and cone-compactness. SIAM J. Appl. Math. 1978; 34: 211–222.

    Article  MathSciNet  MATH  Google Scholar 

  • Henig MI. 1. Proper efficiency with respect to cones. J. Optim. Theory Appl. 1982; 36: 387–407.

    Article  MathSciNet  MATH  Google Scholar 

  • Henig MI. 2. Existence and characterization of efficient decisions with respect to cones. Math. Programming. 1982; 23: 111–116.

    Article  MathSciNet  MATH  Google Scholar 

  • Hurwicz L. ‘Programming in Linear Spaces’.- In: Studies in Linear and Nonlinear Programming, K.J. Arrow, L. Hurwicz and H. Uzawa, eds., Stanford: Standford University Press, 1958.

    Google Scholar 

  • Hyers, D.H., Isac, G. and Rassias, Th.M. Topics in Nonlinear Analysis and Applications. New Jersey-London-Hong Kong: World Scientific, Singapore, 1997.

    Book  MATH  Google Scholar 

  • Isac, G. 1. Cônes Localment Bornés et Cônes Complètement Réguliers. Application a l’Analyse Nonlinéare. Seminaire d’Analyse Moderne Nr.17, Université de Sherbrooke, 1980.

    Google Scholar 

  • Isac, G. 2. The (M)-(L) type duality for locally convex lattices. Rev. Roumaine Math. Pures Appl. 1971; 16: 217–223.

    MathSciNet  MATH  Google Scholar 

  • Isac, G. 3. Sur l’éxistence de l’optimum de Pareto. Rèv. Mat. Univ. Parma. 1983; (4) 9: 303–325.

    MathSciNet  Google Scholar 

  • Isac, G. 4. Un critère de sommabilité absolue dans les espaces localement con vexes ordonnés: cônes nucléaires. Mathematica. 1983; 25(48): 159–169.

    MathSciNet  Google Scholar 

  • Isac, G. 5. Supernormal cones and absoulte summability. Libertas Math. 1985; 5: 17–31.

    MathSciNet  MATH  Google Scholar 

  • Isac, G. 6. Supernormal cones and fixed point theory. Rocky Mountain J. Math. 1987; 17: 219–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Isac, G. 7. Pareto optimization in infinite dimensional spaces: the importance of nuclear cones. J. Math. Anal. Appl. 1994; 182: 393–404.

    Article  MathSciNet  MATH  Google Scholar 

  • Isac, G. 8. ‘The Ekeland’s principle and Pareto -efficiency’. — In: Multi-Objective Programming and Goal Programming, M. Tamiz, ed., Lecture Notes in Econom. Math. Systems Nr.432, Springer-Verlag, 1996. — P. 148–162.

    Chapter  Google Scholar 

  • Isac, G. 9. Ekeland’s principle and nuclear cones: a geometrical aspect. Math. Comput. Modelling. 1997; 26: 111–116.

    Article  MathSciNet  MATH  Google Scholar 

  • Isac, G. 10. “On Pareto efficiency. A general constructive existence principle.” Preprint, 1998.

    Google Scholar 

  • Isac, G. and Postolica, V. The Best Approximation and Optimization in Locally Convex Spaces, Verlag Peter Lang: Approximation and Optimization, Vol.2, Frankfurt am Main, 1993.

    MATH  Google Scholar 

  • Isermann H. Proper efficiency and the linear vector maximum problem. Operations Res. 1974; 22: 189–191.

    Article  MathSciNet  MATH  Google Scholar 

  • Jahn J. 1. Existence theorems in vector optimization. J. Optim. Theory Appl. 1986; 50: 397–406.

    Article  MathSciNet  MATH  Google Scholar 

  • Jahn J. 2. Mathematical Vector Optimization in Partially Ordered Linear Spaces, Frankfurt: Peter Lang, 1986.

    MATH  Google Scholar 

  • Jahn J. 3. A generalization of a theorem of Arrow, Barankin and Blackwell. SLAM J. Control and Opt. 1988; 26: 999–1005.

    Article  MathSciNet  MATH  Google Scholar 

  • Jameson, G.O. Ordered Linear Spaces, Springer-Verlag, Lecture Notes in Mathematics Nr.141, 1970.

    MATH  Google Scholar 

  • Karlin S. Positive operators. J. Math. Mech. 1959; 8: 907–937.

    MathSciNet  MATH  Google Scholar 

  • Klinger A. Improper solutions of the vector maximum problem. Operations Research. 1967; 15: 570–572.

    Article  MATH  Google Scholar 

  • Krasnoselskii, M.A., Lifshits, Je.A. and Sobolev, A.V. Positive Linear Systems. The Method of Positive Operators. Berlin: Heldermann Verlag, 1989.

    Google Scholar 

  • Krein MG and Rutman MA. Linear operators which leave a cone in a Banach space invariant. Uspeki Mat. Nauk. 1948; 3: 3–95.

    MathSciNet  MATH  Google Scholar 

  • Kuhn HW and Tucker AW. ‘Nonlinear Programming’.- In: Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman, ed., University of California Press, Berkeley, Ca, 1950. P. 481–492.

    Google Scholar 

  • Loridan P. -solution in vector minimization problems. J. Opt. Theory Appl. 1984; 42: 265–276.

    Article  MathSciNet  Google Scholar 

  • Luc DT. 1. An existence theorem in vector optimization. Math. Oper. Res. 1989; 14: 693–699.

    Article  MathSciNet  MATH  Google Scholar 

  • Luc DT. 2. Contractibility of efficient point sets in normed spaces. Nonlinear Anal. Theory Math. Appl. 1990; 15: 527–535.

    Article  MATH  Google Scholar 

  • Luc DT. 3. Recession cones and the domination property in vector optimization. Math. Programming. 1991; 49: 113–122.

    Article  Google Scholar 

  • Luc DT. 4. Theory of Vector Optimization, Lecture Notes in Economics and Math. Systems, Vol. 319. New York — Berlin: Springer-Verlag, 1989.

    Book  Google Scholar 

  • Luc DT. 5. On the domination property in vector optimization. J. Opt. Theory Appl. 1984; 43: 327–330.

    Article  MathSciNet  MATH  Google Scholar 

  • Luc DT. 6. Connectedness of the efficient point sets in quasiconcave vector maximization. J. Math. Anal. Appl. 1987; 12: 346–354.

    Article  MathSciNet  Google Scholar 

  • Luc DT. 7. Structure of the efficient point set. Proc. Amer. Math. Soc. 1985; 19: 433–440.

    Article  Google Scholar 

  • Luc DT. 8. Scalarization of vector optimization problems. J. Optim. Theory Appl. 1987; 55: 85–102.

    Article  MathSciNet  MATH  Google Scholar 

  • Majundar M. Some general theorems on efficiency prices with an infinite dimensional commodity space. J. Econom. Theory. 1972; 5: 1–13.

    Article  MathSciNet  Google Scholar 

  • Makarov EK and Rachkovski NN. Density theorems for generalized Henig proper efficiency. J. Optim. Theory Appl. 1996; 91: 419–437.

    Article  MathSciNet  MATH  Google Scholar 

  • Marinescu, Gh. Espaces Pseudotopologiques et Thiorie des Distributions. Berlin: D.V.W., 1963.

    Google Scholar 

  • Molho, E. and Zaffaroni, A. ‘Quasiconvexity of sets and connectedness of the efficient frontier in ordered vector spaces’.- In: Generalized Convexity, Generalized Monotonicity: Recent Results, J.P. Crouzeix et al., eds., Dordrecht: Kluwer Academic Publishers, 1998.- P. 408–424.

    Google Scholar 

  • Mokobodski, G. 1. ‘Cônes normaux et éspaces nucléaires: Cônes semi-complets’. — In: Seminaire Choquet, Iniation a l’Analyse 7-ème Année, 1967/1968, B-6.

    Google Scholar 

  • Mokobodski, G. 2. ‘Principe de Balayage, principe de domination’. In: Seminaire de Choquet 1, 1967/1968, B-6.

    Google Scholar 

  • Morozov VV. On properties of the set of nondominated vectors. Vestnik Moskov. Univ. Comput. Sci. Cyber. 1977; 4: 51–61.

    Google Scholar 

  • Naccache PH. Connectedness of the set of nondominated outcomes in multicriteria optimization. J. Optim. Theory Appl. 1978; 25: 459–467.

    Article  MathSciNet  MATH  Google Scholar 

  • Nachbin, L. Topology and Order. New York: Van Nostrand, 1965.

    MATH  Google Scholar 

  • Németh AB. Between Pareto efficiency and Pareto -efficiency. Optimization. 1989; 20: 615–637.

    Article  MathSciNet  MATH  Google Scholar 

  • Pareto, V. 1. Manuale di Economia Politica. Società Editrice Libraria, Milano, Italy (1906); Piccola Biblioteca Scientifica Nr.13, Societa Editrice Libraria Milano, Italy, 1919.

    Google Scholar 

  • Pareto, V. 2. Sociological Writings. Selected and introduced by S. E. Finer, Translated by D. Mirfin, Frederick A. Praeger, New York, New York, 1966.

    Google Scholar 

  • Peleg B. Topological properties of the efficient points set. Proc. Amer. Math. Soc. 1972; 35: 531–536.

    Article  MathSciNet  MATH  Google Scholar 

  • Penot JP. L’optimization à la Pareto: Deux ou trois chose que je sais d’elle. Publ. Math. Univ. Pan, 1978.

    Google Scholar 

  • Peressini, A.L. Ordered Topological Vector Spaces. New York: Harper and Row Publisher, 1967.

    MATH  Google Scholar 

  • Petschke M. On a theorem of Arrow, Barankin and Blackwell. SIAM J. Control Opt. 1990; 28: 395–401.

    Article  MathSciNet  MATH  Google Scholar 

  • Pontini C. Inclusion theorems for non-explosive and strongly exposed cones in normed spaces. J. Math. Anal. Appl. 1990; 148: 275–286.

    Article  MathSciNet  MATH  Google Scholar 

  • Popovici N. ‘Contribution à l’Optimization Vectorielle’, Thèse, Université de Limoges, France, 1995.

    Google Scholar 

  • Postolica V. 1. Vectorial optimization programs with multifunctions and duality. Ann. Sci. Math. Quebec 1986; 10: 85–102.

    MathSciNet  MATH  Google Scholar 

  • Postolica V. 2. ‘Some existence results concerning the efficient points in locally convex spaces’. — In: Babes-Bolyai Univ. Faculty of Math., Seminar on Math. Analysis, 1987. -P. 715–80.

    Google Scholar 

  • Postolica V. 3. ‘Existence results for the efficient points in locally convex spaces ordered by supernormal cones and con ically bounded sets’. -In: Babes-Bolyai Univ. Faculty of Math., Seminar on Math. Anal., 1988. — P. 187–192.

    Google Scholar 

  • Postolica V. 4. Existence conditions of efficient points for multifunctions with values in locally convex spaces. Stud. Cere. Mat. 1989; 41: 325–331.

    MathSciNet  MATH  Google Scholar 

  • Postolica V. 5. New existence results for efficient points in locally convex spaces ordered by supernormal cones. J. Global Optim. 1993; 3: 233–242.

    Article  MathSciNet  MATH  Google Scholar 

  • Postolica V. 6. Properties of Pareto sets in locally convex spaces. Optimization 1995; 34: 223–229.

    Article  MathSciNet  MATH  Google Scholar 

  • Postolica V. 7. An extension to sets of supernormal cones and generalized subdifferential. Optimization. 1994; 29: 131–139.

    Article  MathSciNet  MATH  Google Scholar 

  • Postolica V. 8. “Recent conditions for Pareto efficiency in locally convex spaces”. Preprint, 1997.

    Google Scholar 

  • Preocupanu Th. Scalar minimax properties in vectorial optimization. International Series of Numerical Mathematics. 1992; 107: 299–306.

    Google Scholar 

  • Radner, R. ‘A note on maximal points of convex sets in l ’. — In: Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 1967. — P. 351–354.

    Google Scholar 

  • Rudin, W. Functional Analysis. New York: McGraw-Hill, 1973.

    MATH  Google Scholar 

  • Salz W. Eine topologische Eigenschaft der effizienten Punkte konvexer Mengen. Operat. Res. Verfahren. 1976; 23: 197–203.

    Google Scholar 

  • Sawaragi, Y., Nakayama, H. and Tanino, T. Theory of Multiobjective Optimisation. Academic Press, 1985.

    Google Scholar 

  • SchafFer, H.H. Topological Vector Spaces. New York: MacMillan Company, 1966.

    Google Scholar 

  • Song W. Generalization of the Arrow-Barankin-Blackwell theorem in a dual space setting. J. Optim. Theory Appl. 1997; 95: 225–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Stadler W. A survey of multicriteria optimization or the vector maximum problem. Part (I): 1776–1960. J. Optim. Theory Appl. 1979; 29: 1–52.

    Article  MathSciNet  MATH  Google Scholar 

  • Staib T. On two generalization of Pareto minimality. J. Optim. Theory Appl. 1988; 59: 289–306.

    MathSciNet  MATH  Google Scholar 

  • Sterna-Karwat A. 1. On existence of cone-maximal points in a real topological linear spacte. Israel J. Math. 1986; 54: 33–41.

    Article  MathSciNet  MATH  Google Scholar 

  • Sterna-Karwat A. 2. ‘On the existence of cone-efficient points’. — In: Recent Advances and Historical Development of Vector Optimization, J. Jahn and W. Krabs, eds., Lecture Notes in Economics and Mathematical Systems, Nr. 294, Springer-Verlag, 1986. — P. 233–240.

    Google Scholar 

  • Tammer, Chr. 1. ‘Existence results and necessary conditions for -efficient elements’. -In: Multicriteria Decision-Proceedings of the 14-th Meeting of the German Working Group “Mehrhriterielle Entsch”, Brosowski e.a., eds., Frankfurt/Main-Bern: Lang-Verlag, 1993. — P. 97–110.

    Google Scholar 

  • Tammer, Chr. 2. “A variational principle and applications for vectorial control approximation problems”. Report Nr.9, 1996, Martin-Luher-Universitat, Halle-Wittenberg, Germany.

    Google Scholar 

  • Tammer, Chr. 3. “Approximate solutions of vector-valued control-approximation problems”. Report Nr.27, 1996, Martin-Luher-Universitat, Halle-Wittenberg, Germany.

    Google Scholar 

  • Tang Y. Conditions for constrained efficient solutions of multiobjective problems in Banach spaces. J. Math. Anal. Appl. 1983; 96: 505–519.

    Article  MathSciNet  MATH  Google Scholar 

  • Trevis, F. Locally Convex Spaces and Linear Partial Differential Equations. New York: Springer-Verlag, 1967.

    Book  Google Scholar 

  • Truong XDH. 1. On the existence of efficient points in locally convex spaces. J. Global Optim. 1994; 4: 265–278.

    Article  MathSciNet  MATH  Google Scholar 

  • Truong XDH. 2. A note on a class of cones ensuring the existence of efficient points in bounded complete sets. Optimization. 1994; 31: 141–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Truong XDH. 3. Cones admitting strictly positive functionals and scalarization of some vector optimization problems. J. Optim. Theory Appl. 1997; 93: 355–372.

    Article  MathSciNet  MATH  Google Scholar 

  • Truong XDH. 4. “Existence and density results for proper efficiency in cone compact sets”. Preprint, Hanoi Institute of Mathematics, 1999.

    Google Scholar 

  • Valyi, I. ‘Approximate solutions of vector optimization problems’. — In: Systems Analysis and Simulation, A. Sydow, M. Thoma and R. Vychnevetski, eds., Berlin DDR: Akademic-Verlag, Vol. 1, 1985. — P. 246–250.

    Google Scholar 

  • Varaiya PP. Nonlinear programming in Banach spaces. SIAM J. Appl. Math. 1967; 15: 284–293.

    Article  MathSciNet  MATH  Google Scholar 

  • Vogel, W. ‘Vektoroptimierung in Produktraumen’, Math. Systems in Economics 35, Verlag Anton Hain, Meisenheim am Glan, 1977.

    Google Scholar 

  • Wagner DH, Semi-compactness with respect to an Euclidean cone. Canad. J. Math. 1997; 29: 29–36.

    Article  Google Scholar 

  • Wantao F. On the density of proper efficient points. Proc. Amer. Math. Soc. 1996; 124: 1213–1217.

    Article  MathSciNet  MATH  Google Scholar 

  • Warburton AR. Quasiconcave vector maximization: connectedness of the sets of Pareto optimal and weak Pareto optimal alternatives. J. Optim. Theory Appl. 1983; 40: 537–557.

    Article  MathSciNet  MATH  Google Scholar 

  • Wong, Y.Ch. Schwartz Spaces, Nuclear Spaces and Tensor Product, Lecture Notes in Math. Vol. 726. New York — Berlin: Springer-Verlag, 1979.

    Google Scholar 

  • Yu PL. Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjective. J. Optim. Theory Appl. 1974; 14: 319–377.

    Article  MATH  Google Scholar 

  • Zheng XY. 1. Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 1997; 94: 469–486.

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng XY. 2. The domination property for efficiency in locally convex topological vector spaces. J. Math. Anal. Appl. 1997; 213: 455–467.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhuang DM. 1. Bases of convex cones and Borwein’s proper efficiency. J. Optim. Theory Appl. 1991; 71: 613–620.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhuang DM. 2. Density results for proper efficiency. SIAM J. Control Opt. 1994; 32: 51–58.

    Article  MATH  Google Scholar 

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Isac, G., Bulavsky, V.A., Kalashnikov, V.V. (2002). Efficiency. In: Complementarity, Equilibrium, Efficiency and Economics. Nonconvex Optimization and Its Applications, vol 63. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3623-6_12

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  • DOI: https://doi.org/10.1007/978-1-4757-3623-6_12

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