Abstract
The present note is concerned with the study of the relations between the notions of asymptotic cones introduced by Dedieu and that of recession cones introduced by Luc. Conditions under which these notions coincide are given, as well as the fact that the compactness condition used by Luc is related (more restrictively) to asymptotic compactness. As an application of these notions, a result on proper efficiency in the sense of Lampe, established by Luc in finite dimensions, is extended to the infinite-dimensional case.
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References
Debreu, G.,Theory of Value, Yale University Press, New Haven, Connecticut, 1975.
Dedieu, J. P., Cône Asymptote d'un Ensemble Non Convexe: Application à l'Optimisation, Comptes Rendus des Séances de l'Académie des Sciences, Série I, Mathématique, Vol. 285, pp. 501–503, 1977.
Luc, D. T.,Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, Germany, Vol. 319, 1989.
Luc, D. T.,Recession Cones and the Domination Property in Vector Optimization, Mathematical Programming, Vol. 49, pp. 113–122, 1990.
Dedieu, J. P., Critères de Fermeture pour l'Image d'un Fermé Non Convexe par une Multiapplication, Comptes Rendus des Séances de l'Académie des Sciences, Série I, Mathématique, Vol. 287, pp. 941–943, 1978.
Gwinner, J.,Closed Images of Convex Multivalued Mappings in Linear Topological Spaces with Applications, Journal of Mathematical Analysis and Applications, Vol. 60, pp. 75–86, 1977.
Goossens, P.,Asymptotically Compact Sets, Asymptotic Cone, and Closed Conical Hull, Bulletin de la Societé Royale des Sciences de Liège, Vol. 53, pp. 57–67, 1984.
ZĂlinescu, C., Stabilitè pour une Classe de Problèmes de Optimisation Non Convexes, Comptes Rendus des Séances de l'Académie des Sciences, Série I, Mathématique, Vol. 307, pp. 643–646, 1988.
ZĂlinescu, C.,Stability for a Class of Nonlinear Optimization Problems and Applications, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. F. Dem'yanov, and F. Giannessi, Plenum Press, New York, New York, pp. 437–458, 1989.
Precupanu, T.,Topological Linear Spaces, University of Iasi, Iasi, Romania, 1986 (in Rumanian).
Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, Berlin, Germany, 1975.
ZĂlinescu, C.,On Two Notions of Proper Efficiency, Optimization in Mathematical Physics, Edited by B. Brosowski and E. Martensesn, Methoden und Verfahren der Mathematischen Physik, Vol. 34, pp. 77–86, 1987.
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Communicated by F. Giannessi
This work was partially done while the author was visiting the Department of Mathematics of the University of Pisa, Pisa, Italy.
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ZĂlinescu, C. Recession cones and asymptotically compact sets. J Optim Theory Appl 77, 209–220 (1993). https://doi.org/10.1007/BF00940787
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DOI: https://doi.org/10.1007/BF00940787