×
Gong, X. H.

Connectedness of efficient solution sets for set-valued maps in normed spaces. (English) Zbl 0845.90104

J. Optimization Theory Appl. 83, No. 1, 83-96 (1994).
A new interesting step is reported to find out conditions for vector optimization problems such that the connectedness of efficient sets follows. Here, the density theorem of Arrow-Barankin and Blackwell [look at Contribution to the Theory of Games, H. W. Kuhn, A. W. Tucker (eds.), Princeton University Press, 87-92 (1953; Zbl 0041.25302)] has been generalized so that it could be applied to vector optimization problems with point-to-set functions \(f: A\to Y\), where \(A\) is a set in a Hausdorff space \(X\) and \(Y\) is a real normed space.
Reviewer: A.Göpfert (Halle)

Cited in 2 Reviews
Cited in 38 Documents

MSC:

90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
54C60 Set-valued maps in general topology

Keywords:

vector optimization; connectedness of efficient sets; theorem of Arrow-Barankin and Blackwell; point-to-set functions

Citations:

Zbl 0041.25302
PDF BibTeX XML Cite
\textit{X. H. Gong}, J. Optim. Theory Appl. 83, No. 1, 83--96 (1994; Zbl 0845.90104)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Giannessi, F.,Theorems of the Alternative for Multifunctions with Applications to Optimization: General Results, Journal of Optimization Theory and Applications, Vol. 55, pp. 233–256, 1987. · Zbl 0622.90084
[2] Ferrero, O.,Theorems of the Alternative for Set-Valued Functions in Infinite-Dimensional Spaces, Optimization, Vol. 20, pp. 167–175, 1989. · Zbl 0676.90099
[3] Mei J. L.,Theorem of the Alternative for a Cone-Convex Set-Valued Mapping, Applied Mathematics: Journal of Chinese Universities, Vol. 7, pp. 54–63, 1992. · Zbl 0765.52012
[4] Jeyakumar, V.,A General Farkas Lemma and Characterization of Optimality for a Nonsmooth Program Involving Convex Processes, Journal of Optimization Theory and Applications, Vol. 55, pp. 449–461, 1987. · Zbl 0616.90072
[5] Corley, H. W.,Existence and Lagrangian Duality for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987. · Zbl 0595.90085
[6] Corley, H. W.,Optimality Conditions for Maximizations of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 58, pp. 1–10, 1988. · Zbl 0621.90083
[7] Warburton, A. R.,Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537–557, 1983. · Zbl 0496.90073
[8] Luc, D. T.,Contractibility of Efficient Point Sets in Normed Spaces, Nonlinear Analysis, Vol. 15, pp. 527–535, 1990. · Zbl 0718.90077
[9] Jahn, J.,Mathematical Vector Optimiziation in Partially-Ordered Linear Spaces, Peter Lang, Frankfurt/Main, Germany, 1986. · Zbl 0578.90048
[10] Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974. · Zbl 0268.90057
[11] Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contributions to the Theory of Games, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 87–92, 1953.
[12] Hartley, R.,On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978. · Zbl 0379.90005
[13] Bitran, G. R., andMagnanti, T. L.,The Structure of Admissible Points with Respect to Cone Dominance, Journal of Optimization Theory and Applications, Vol. 29, pp. 573–614, 1979. · Zbl 0389.52021
[14] Borwein, J. M.,The Geometry of Pareto Efficiency over Cones, Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 11, pp. 235–248, 1980. · Zbl 0447.90077
[15] Jahn, J.,A Generalization of a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 26, pp.999–1005, 1988. · Zbl 0652.90093
[16] Dauer, J. P., andGallagher, R. J.,Positive Proper Efficient Points and Related Cone Results in Vector Optimization Theory, SIAM Journal on Control and Optimization, Vol. 28, pp. 158–172, 1990. · Zbl 0697.90072
[17] Petschke, M.,On a Theorem of Arrow, Barankin, and Blackwell, SIAM Journal on Control and Optimization, Vol. 28, pp. 395–401, 1990. · Zbl 0692.49005
[18] Fu, W. T.,On a Problem of Arrow-Barankin-Blackwell, OR and Decision Making, Vol. 2, pp. 1164–1169, 1992 (in Chinese).
[19] Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, John Wiley and Sons, New York, New York, 1984. · Zbl 0641.47066
[20] Naccache, P. H.,Connectedness of the Set of Nondominated Outcomes in Multicriteria Optimization, Journal of Optimization Theory and Applications, Vol. 25, pp. 459–467, 1978. · Zbl 0363.90108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.