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Pareto efficiency in locally convex spaces. I. (English) Zbl 0665.49003
A general notions of Pareto efficiency is introduced and we study the algebraic and topological properties of the set of all efficient points. We also define a notion of weak Pareto efficiency and compare it with the initial one. Then we pass to the study of Pareto efficiency in the case where the sets under consideration are random. We obtain characterizations of the elements of the efficiency set and using the theory of set valued integration we characterize the aggregate efficiency set.
[For part II see the author, ibid. 8, 117-136 (1985; Zbl 0665.49004).]

MSC:
49J27 Existence theories for problems in abstract spaces
90C31 Sensitivity, stability, parametric optimization
90C48 Programming in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Arrow K. J., Contributions to the Theory of Games 2 (1953)
[2] Asimow L., Convexity Theory and its Applications to Functional Analysis (1980) · Zbl 0453.46013
[3] Aumann R., J. Math. Anal. Appl. 12 pp 1– (1965) · Zbl 0163.06301 · doi:10.1016/0022-247X(65)90049-1
[4] Benson H. P., J. Optimization Theory Appl. 26 pp 569– (1978) · Zbl 0373.90085 · doi:10.1007/BF00933152
[5] Bitran G. R., J. Optimization Theory Appl. 29 pp 573– (1979) · Zbl 0389.52021 · doi:10.1007/BF00934453
[6] Blackwell, D. and Girshick, M. A. 1954. ”Theory of Games and Statistical Decisions”. New York: Wiley. · Zbl 0056.36303
[7] Borwein J., SIAM J. Control Optimization 15 pp 57– (1977) · Zbl 0369.90096 · doi:10.1137/0315004
[8] Cesari L., Trans. Amer. Math. Soc. 244 pp 37– (1978) · doi:10.1090/S0002-9947-1978-0506609-1
[9] Cesari L., Nonlinear Anal 2 pp 225– (1978) · Zbl 0383.49003 · doi:10.1016/0362-546X(78)90068-8
[10] Clarke F., Adv. Math. 40 pp 52– (1981) · Zbl 0463.49017 · doi:10.1016/0001-8708(81)90032-3
[11] Cornwall R., J. Math. Anal. Appl. 39 pp 771– (1972) · Zbl 0202.33801 · doi:10.1016/0022-247X(72)90197-7
[12] Debreu G., Decision Processes (1954)
[13] Debreu G., Theory of Value (1959) · Zbl 0193.20205
[14] Diestel, J. and Uhl, J. J. 1977. ”Vector Measures, Math. Surveys”. Vol. 15, Providence: A.M.S. · Zbl 0369.46039 · doi:10.1090/surv/015
[15] Dunford, N. and Schwartz, J. 1958. ”Linear Operators”. Vol. 1, New York: Wiley. · Zbl 0084.10402
[16] Floret K., Weakly compact sets 801 (1980) · Zbl 0437.46006 · doi:10.1007/BFb0091483
[17] Geoffrion A. M., J. Math. Anal. Appl. 22 pp 618– (1968) · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[18] Hiai F., J. Mult. Anal. 7 pp 149– (1977) · Zbl 0368.60006 · doi:10.1016/0047-259X(77)90037-9
[19] Hildenbrand, W. 1974. ”Core and Equilibria of a Large Economy”. Princeton: Princeton Univ. Press. · Zbl 0351.90012
[20] Himmelberg C., Fund. Math. pp 53– (1975)
[21] Hiriart-Urruty J. B., Bull. French Math. Soc. 60 pp 57– (1979)
[22] Keeney R., Decisions with Multiple Objectives: Preferences and Value Trade-offs (1976)
[23] Klee V., Proc. Amer. Math. Soc. 6 pp 313– (1955) · doi:10.1090/S0002-9939-1955-0068113-7
[24] Nacchache P. H., J. Optimiation Theory Appl. 25 pp 459– (1978) · Zbl 0363.90108 · doi:10.1007/BF00932907
[25] Nieuwenhuis J. W., J. Optimization Theory Appl. 36 pp 289– (1982) · Zbl 0452.90074 · doi:10.1007/BF00933835
[26] Papageorgiou N. S., Pacific J. Math. 107 (2) pp 403– (1983) · doi:10.2140/pjm.1983.107.403
[27] Papageorgiou N. S., Clarke’s theory, Pacific J. Math. 109 (2) pp 463– (1983) · doi:10.2140/pjm.1983.109.463
[28] Papageorgiou N. S., Stability results 109 (2) (1983)
[29] Peressini A., Ordered Topological Vector Spaces (1967) · Zbl 0169.14801
[30] Rockafellar R. T., Convex Analysis (1976) · Zbl 0333.90008
[31] Rockafellar R. T., J. Math. Anal. Appl. 28 pp 4– (1969) · Zbl 0202.33804 · doi:10.1016/0022-247X(69)90104-8
[32] Rockafellar R. T., Nonlinear Operators and the Calculus of Variations Gossez 543 (1976) · Zbl 0358.90053
[33] Schaefer H., Graduate Texts in Math. 3 (1971)
[34] Smale S., J. Math. Econ. 1 pp 107– (1974) · Zbl 0316.90007 · doi:10.1016/0304-4068(74)90002-0
[35] Valadier M., Math. Scand. 30 pp 65– (1972) · Zbl 0239.46038 · doi:10.7146/math.scand.a-11064
[36] Yu P. L., J. Optimization Theory Appl. 14 pp 319– (1974) · Zbl 0268.90057 · doi:10.1007/BF00932614
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