# Efficiency

• G. Isac
• V. A. Bulavsky
• V. V. Kalashnikov
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 63)

## 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).

## Keywords

Convex Cone Vector Optimization Convex Space Vector Optimization Problem Closed Convex Cone
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 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. 87–92Google Scholar
2. 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
3. Bahya AO. 1. Étude des cônes nucléaires. Ann. Sci. Math. Quebec. 1991; 15: 123–133
4. 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
5. Bahya AO. 3. Résultats sur des cônes séquentiellement complètement réguliers’.Preprint, 1995.Google Scholar
6. 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
7. 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.
8. Benson HP. 1. An improved definition of proper efficiency for vector minimization with respect to cones. J. Math. Anal. Appl. 1979; 71: 232–241.
9. Benson HP. 2. On a domination property for vector maximization with respect to cones. J. Optim. Theory Appl. 1983; 39: 125–132;
10. Benson HP. 2. Errata Corrige, J. Optim. Theory Appl. 1984; 43: 477–479.
11. Benson HP. 3. Efficiency and proper efficiency for vector minimization with respect to cones. J. Math. Anal. Appl. 1983; 93: 273–289.
12. Benson HP and Morin TL. The vector maximization problem: proper efficiency and stability. SIAM J. Appl. Math. 1977; 32: 64–72.
13. 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.
14. Bitran GR and Magnanti TL. The structure of admissible points with respect to cone dominance. J. Optim. Theory Appl. 1979; 29: 573–614.
15. Borwein JM. 1. Proper efficient points for maximization with respect to cones. SIAM J. Control Opt. 1977; 15: 57–63.
16. Borwein JM. 2. The geometry of Pareto efficiency over cones. Math. Operationsforsch. Scr. Optim. 1980; 11: 235–248.
17. Borwein JM. 3. On the existence of Pareto efficient points. Math. Oper. Res. 1983; 8: 68–75.Google Scholar
18. Borwein JM and Zhuang DM. 1. Super-efficiency in vector optimization. ZOR — Methods and Models of Operations Research. 1991; 35: 175–184.
19. Borwein JM and Zhuang DM. 2. Super-efficiency in vector optimization. Trans. Amer. Soc. 1993; 338: 105–122.
20. 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.
21. Cesary L and Suryanarayana MB. Existence theorems for Pareto optimization, multivalued and Banach space valued functional. Trans. Amer. Math. Soc. 1978; 244: 37–65.
22. Chen Guang-Ya. On generalized Arrow-Barankin-Blackwell theorems in locally convex spaces. J. Optim. Theory Appl. (in press)Google Scholar
23. 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.
24. 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.
25. Chew KL. Maximal points with respect to cone dominance in Banach spaces and their existence. J. Optim. Theory Appl. 1984; 44: 1–53.
26. 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.
27. Choo EV and Atkins DR. Connectedness in multiple linear fractional programming. Management Science. 1983; 29: 250–255.
28. Choquet, G. Lectures on Analysis, Vol. 1–3, Mathematics Lecture Notes Series, New York-Amsterdam: Benjamin, 1969.Google Scholar
29. Constantinescu G. L’ensemble des fonctions surharmoniques positives sur un esipace harmonique est faiblement complet. C.R.Acad. Sci. Paris. 1970; 271: A549-A551.
30. Corley HW. 1. An existence result for maximization with respect to cones. J. Optim. Theory Appl. 1980; 31: 277–281.
31. Corley HW. 2. Some hybrid fixed point theorems related to optimization. J. Math. Anal. Appl. 1986; 120: 528–532.
32. 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.
33. 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.
34. Dauer JP and Stadler W. A survey of vector optimization in infinite dimensional spaces. Part II. J. Optim. Theory Appl. 1986; 51: 205–241.
35. Durier R and Michelot C. Sets of efficient points in a normed space. J. Mith. Anal. Appl. 1986; 117: 506–528.
36. Edwards, R.E. Functional Analysis. New York: Holt Reinhart and Winston, 1965.
37. Ekeland I. 1. Sur les problemes variationnels. C.R. Acad. Sci. Paris. 1972; 275: A1057-A1059.
38. Ekeland I. 2. On some variational principle. J. Math. Anal. Appl. 1974; 47: 324–354.
39. Ekeland I. 3. Nonconvex minimization problems. Bull. Amer. Math. Soc. 1979; 1(3): 443–474.
40. Ekeland I. 4. Some lemmas about dynamical systems. Math. Scan. 1983; 52: 262–268.
41. 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.
42. Fan K. Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 1953; 39: 42–47.
43. Ferrot F. 1. A generalization of the Arrow-Barankin-Blackwell theorem in normed spaces. J. Math. Anal. Appl. 1991; 158: 47–54.
44. 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
45. Ferrot F. 3. A new ABB theorem in Banach spaces. Preprint, Universita di Genova, Italy, 1999.Google Scholar
46. Fu WT. On the density of proper efficient points. Proceed. Amer. Math. Soc. 1996; 124: 1213–1217.
47. Gallagher RJ and Saleh OA. Two generalizations of a theorem of Arrow-Barankin-Blackwell. SIAM J. Control Opt. 1993; 31: 247–256.
48. Georffrion AM. Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 1968; 22: 618–630.
49. Gopfert A and Tammer Chr. 1. A new maximal point theorem. ZAA. 1995; 14: 379–390.
50. 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
51. 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
52. 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
53. Gopfert A, Tammer Chr. and Zalinescu C. 2. “Maximal point theorems in product spaces.” Preprint, University of Halle, Germany, 1998.Google Scholar
54. Gopfert A, Sekatzek M and Tammer Chr. “Novelties around Ekeland’s variational principle.” Preprint, University of Halle, Germany, 1999.Google Scholar
55. Gong XH. Connectedness of efficient solution set for set-valued maps in normed spaces. J. Optim. Theory Appl. 1994; 83: 83–96.
56. Gotz A and Jahn J. The Lagrange multiplier rule in set-valued optimization. (To appear in SIAM J. Opt.)Google Scholar
57. Guerraggio A, Molho E and Zaffaroni A. On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 1994; 82: 1–21.
58. Hartley R. On cone-efficiency, cone-convexity and cone-compactness. SIAM J. Appl. Math. 1978; 34: 211–222.
59. Henig MI. 1. Proper efficiency with respect to cones. J. Optim. Theory Appl. 1982; 36: 387–407.
60. Henig MI. 2. Existence and characterization of efficient decisions with respect to cones. Math. Programming. 1982; 23: 111–116.
61. 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
62. Hyers, D.H., Isac, G. and Rassias, Th.M. Topics in Nonlinear Analysis and Applications. New Jersey-London-Hong Kong: World Scientific, Singapore, 1997.
63. 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
64. Isac, G. 2. The (M)-(L) type duality for locally convex lattices. Rev. Roumaine Math. Pures Appl. 1971; 16: 217–223.
65. Isac, G. 3. Sur l’éxistence de l’optimum de Pareto. Rèv. Mat. Univ. Parma. 1983; (4) 9: 303–325.
66. Isac, G. 4. Un critère de sommabilité absolue dans les espaces localement con vexes ordonnés: cônes nucléaires. Mathematica. 1983; 25(48): 159–169.
67. Isac, G. 5. Supernormal cones and absoulte summability. Libertas Math. 1985; 5: 17–31.
68. Isac, G. 6. Supernormal cones and fixed point theory. Rocky Mountain J. Math. 1987; 17: 219–226.
69. Isac, G. 7. Pareto optimization in infinite dimensional spaces: the importance of nuclear cones. J. Math. Anal. Appl. 1994; 182: 393–404.
70. 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.
71. Isac, G. 9. Ekeland’s principle and nuclear cones: a geometrical aspect. Math. Comput. Modelling. 1997; 26: 111–116.
72. Isac, G. 10. “On Pareto efficiency. A general constructive existence principle.” Preprint, 1998.Google Scholar
73. 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.
74. Isermann H. Proper efficiency and the linear vector maximum problem. Operations Res. 1974; 22: 189–191.
75. Jahn J. 1. Existence theorems in vector optimization. J. Optim. Theory Appl. 1986; 50: 397–406.
76. Jahn J. 2. Mathematical Vector Optimization in Partially Ordered Linear Spaces, Frankfurt: Peter Lang, 1986.
77. Jahn J. 3. A generalization of a theorem of Arrow, Barankin and Blackwell. SLAM J. Control and Opt. 1988; 26: 999–1005.
78. Jameson, G.O. Ordered Linear Spaces, Springer-Verlag, Lecture Notes in Mathematics Nr.141, 1970.
79. Karlin S. Positive operators. J. Math. Mech. 1959; 8: 907–937.
80. Klinger A. Improper solutions of the vector maximum problem. Operations Research. 1967; 15: 570–572.
81. Krasnoselskii, M.A., Lifshits, Je.A. and Sobolev, A.V. Positive Linear Systems. The Method of Positive Operators. Berlin: Heldermann Verlag, 1989.Google Scholar
82. Krein MG and Rutman MA. Linear operators which leave a cone in a Banach space invariant. Uspeki Mat. Nauk. 1948; 3: 3–95.
83. 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
84. Loridan P. -solution in vector minimization problems. J. Opt. Theory Appl. 1984; 42: 265–276.
85. Luc DT. 1. An existence theorem in vector optimization. Math. Oper. Res. 1989; 14: 693–699.
86. Luc DT. 2. Contractibility of efficient point sets in normed spaces. Nonlinear Anal. Theory Math. Appl. 1990; 15: 527–535.
87. Luc DT. 3. Recession cones and the domination property in vector optimization. Math. Programming. 1991; 49: 113–122.
88. Luc DT. 4. Theory of Vector Optimization, Lecture Notes in Economics and Math. Systems, Vol. 319. New York — Berlin: Springer-Verlag, 1989.
89. Luc DT. 5. On the domination property in vector optimization. J. Opt. Theory Appl. 1984; 43: 327–330.
90. Luc DT. 6. Connectedness of the efficient point sets in quasiconcave vector maximization. J. Math. Anal. Appl. 1987; 12: 346–354.
91. Luc DT. 7. Structure of the efficient point set. Proc. Amer. Math. Soc. 1985; 19: 433–440.
92. Luc DT. 8. Scalarization of vector optimization problems. J. Optim. Theory Appl. 1987; 55: 85–102.
93. Majundar M. Some general theorems on efficiency prices with an infinite dimensional commodity space. J. Econom. Theory. 1972; 5: 1–13.
94. Makarov EK and Rachkovski NN. Density theorems for generalized Henig proper efficiency. J. Optim. Theory Appl. 1996; 91: 419–437.
95. Marinescu, Gh. Espaces Pseudotopologiques et Thiorie des Distributions. Berlin: D.V.W., 1963.Google Scholar
96. 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
97. 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
98. Mokobodski, G. 2. ‘Principe de Balayage, principe de domination’. In: Seminaire de Choquet 1, 1967/1968, B-6.Google Scholar
99. Morozov VV. On properties of the set of nondominated vectors. Vestnik Moskov. Univ. Comput. Sci. Cyber. 1977; 4: 51–61.Google Scholar
100. Naccache PH. Connectedness of the set of nondominated outcomes in multicriteria optimization. J. Optim. Theory Appl. 1978; 25: 459–467.
101. Nachbin, L. Topology and Order. New York: Van Nostrand, 1965.
102. Németh AB. Between Pareto efficiency and Pareto -efficiency. Optimization. 1989; 20: 615–637.
103. 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
104. 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
105. Peleg B. Topological properties of the efficient points set. Proc. Amer. Math. Soc. 1972; 35: 531–536.
106. Penot JP. L’optimization à la Pareto: Deux ou trois chose que je sais d’elle. Publ. Math. Univ. Pan, 1978.Google Scholar
107. Peressini, A.L. Ordered Topological Vector Spaces. New York: Harper and Row Publisher, 1967.
108. Petschke M. On a theorem of Arrow, Barankin and Blackwell. SIAM J. Control Opt. 1990; 28: 395–401.
109. Pontini C. Inclusion theorems for non-explosive and strongly exposed cones in normed spaces. J. Math. Anal. Appl. 1990; 148: 275–286.
110. Popovici N. ‘Contribution à l’Optimization Vectorielle’, Thèse, Université de Limoges, France, 1995.Google Scholar
111. Postolica V. 1. Vectorial optimization programs with multifunctions and duality. Ann. Sci. Math. Quebec 1986; 10: 85–102.
112. 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
113. 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
114. Postolica V. 4. Existence conditions of efficient points for multifunctions with values in locally convex spaces. Stud. Cere. Mat. 1989; 41: 325–331.
115. Postolica V. 5. New existence results for efficient points in locally convex spaces ordered by supernormal cones. J. Global Optim. 1993; 3: 233–242.
116. Postolica V. 6. Properties of Pareto sets in locally convex spaces. Optimization 1995; 34: 223–229.
117. Postolica V. 7. An extension to sets of supernormal cones and generalized subdifferential. Optimization. 1994; 29: 131–139.
118. Postolica V. 8. “Recent conditions for Pareto efficiency in locally convex spaces”. Preprint, 1997.Google Scholar
119. Preocupanu Th. Scalar minimax properties in vectorial optimization. International Series of Numerical Mathematics. 1992; 107: 299–306.Google Scholar
120. 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
121. Rudin, W. Functional Analysis. New York: McGraw-Hill, 1973.
122. Salz W. Eine topologische Eigenschaft der effizienten Punkte konvexer Mengen. Operat. Res. Verfahren. 1976; 23: 197–203.Google Scholar
123. Sawaragi, Y., Nakayama, H. and Tanino, T. Theory of Multiobjective Optimisation. Academic Press, 1985.Google Scholar
124. SchafFer, H.H. Topological Vector Spaces. New York: MacMillan Company, 1966.Google Scholar
125. Song W. Generalization of the Arrow-Barankin-Blackwell theorem in a dual space setting. J. Optim. Theory Appl. 1997; 95: 225–230.
126. Stadler W. A survey of multicriteria optimization or the vector maximum problem. Part (I): 1776–1960. J. Optim. Theory Appl. 1979; 29: 1–52.
127. Staib T. On two generalization of Pareto minimality. J. Optim. Theory Appl. 1988; 59: 289–306.
128. Sterna-Karwat A. 1. On existence of cone-maximal points in a real topological linear spacte. Israel J. Math. 1986; 54: 33–41.
129. 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
130. 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
131. 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
132. Tammer, Chr. 3. “Approximate solutions of vector-valued control-approximation problems”. Report Nr.27, 1996, Martin-Luher-Universitat, Halle-Wittenberg, Germany.Google Scholar
133. Tang Y. Conditions for constrained efficient solutions of multiobjective problems in Banach spaces. J. Math. Anal. Appl. 1983; 96: 505–519.
134. Trevis, F. Locally Convex Spaces and Linear Partial Differential Equations. New York: Springer-Verlag, 1967.
135. Truong XDH. 1. On the existence of efficient points in locally convex spaces. J. Global Optim. 1994; 4: 265–278.
136. 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.
137. Truong XDH. 3. Cones admitting strictly positive functionals and scalarization of some vector optimization problems. J. Optim. Theory Appl. 1997; 93: 355–372.
138. Truong XDH. 4. “Existence and density results for proper efficiency in cone compact sets”. Preprint, Hanoi Institute of Mathematics, 1999.Google Scholar
139. 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
140. Varaiya PP. Nonlinear programming in Banach spaces. SIAM J. Appl. Math. 1967; 15: 284–293.
141. Vogel, W. ‘Vektoroptimierung in Produktraumen’, Math. Systems in Economics 35, Verlag Anton Hain, Meisenheim am Glan, 1977.Google Scholar
142. Wagner DH, Semi-compactness with respect to an Euclidean cone. Canad. J. Math. 1997; 29: 29–36.
143. Wantao F. On the density of proper efficient points. Proc. Amer. Math. Soc. 1996; 124: 1213–1217.
144. 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.
145. Wong, Y.Ch. Schwartz Spaces, Nuclear Spaces and Tensor Product, Lecture Notes in Math. Vol. 726. New York — Berlin: Springer-Verlag, 1979.Google Scholar
146. Yu PL. Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjective. J. Optim. Theory Appl. 1974; 14: 319–377.
147. Zheng XY. 1. Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 1997; 94: 469–486.
148. Zheng XY. 2. The domination property for efficiency in locally convex topological vector spaces. J. Math. Anal. Appl. 1997; 213: 455–467.
149. Zhuang DM. 1. Bases of convex cones and Borwein’s proper efficiency. J. Optim. Theory Appl. 1991; 71: 613–620.
150. Zhuang DM. 2. Density results for proper efficiency. SIAM J. Control Opt. 1994; 32: 51–58.