Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T05:31:46.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Invex optimisation problems

Published online by Cambridge University Press:  17 April 2009

D.T. Luc  and
C. Malivert
D.T. Luc
Affiliation:
Department of Mathematics, Faculty of Science University of Limoges Limoges, Cedex 87050, France
C. Malivert
Affiliation:
Department of Mathematics, Faculty of Science University of Limoges Limoges, Cedex 87050, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we extend the concept of invexity to set-valued maps and study vector optimisation problems with invex set-valued data. Necessary and sufficient optimality conditions are established in terms of contingent derivatives. Wolfe type dual problems are constructed via two recently developed approaches which guarantee the zero-gap duality property.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 47 - 66
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Aubin, J.P. and Frankowska, H., Set-valued analysis (Birkhaūser, 1990).Google Scholar
[2]Ben-Israel, A. and Mond, B., ‘What is invexity?’, J. Austral. Math. Soc. (Series B) 28 (1986), 1–9.CrossRefGoogle Scholar
[3]Craven, B.D., ‘Invex functions and constrained local minima’, Bull. Austral. Math. Soc. 25 (1981), 37–46.Google Scholar
[4]Craven, B.D., ‘A modified Wolfe dual for weak vector minimization’, Numer. Funct. Anal. Optim. 10 (1989), 899–907.CrossRefGoogle Scholar
[5]Craven, B.D., ‘Nonsmooth multiobjective programming’, Numer. Funct. Anal. Optim. 10 (1989), 49–64.CrossRefGoogle Scholar
[6]Craven, B.D. and Glover, B.M., ‘Invex functions and duality’, J. Austral. Math. Soc. (Series A) 39 (1985), 1–20.CrossRefGoogle Scholar
[7]Dolecki, S. and Malivert, C., ‘General duality for vector optimization’, (submitted).Google Scholar
[8]Egudo, R.R. and Hanson, M.A., ‘Multiobjective duality with invexity’, J. Math. Anal. Appl. 126 (1987), 469–477.CrossRefGoogle Scholar
[9]Hanson, M.A., ‘On sufficiency of the Kuhn-Tucker conditions’, J. Math. Anal. Appl. 80 (1981), 545–550.CrossRefGoogle Scholar
[10]Jahn, J., Mathematical vector optimization in partially ordered linear spaces (Peter Lang, Frankfurt, 1986).Google Scholar
[11]Luc, D.T., Theory of vector optimization: Lecture Notes in Economics and Mathematical Systems 319 (Springer-Verlag, Berlin, Heidelberg, New York, 1989).CrossRefGoogle Scholar
[12]Luc, D.T., ‘Contingent derivatives of set-valued maps and applications to vector optimization’, Math. Programming 50 (1991), 99–111.CrossRefGoogle Scholar
[13]Luc, D.T. and Jahn, J., ‘Axiomatic approach to duality in optimization’, (submitted).Google Scholar
[14]Malivert, C., Contributions à l'optimisation vectorielle, Thèse Université de Limoges, 1990.Google Scholar
[15]Penot, J.-P., ‘Differentiability of relations and differential stability of perturbed optimization problems’, SIAM J. Control Optim. 22 (1984), 529–551.CrossRefGoogle Scholar
[16]Tanaka, Y., ‘Note on generalized convex functions’, J. Optim. Throry Appl. 66 (1990), 345–349.CrossRefGoogle Scholar
[17]Weir, T. and Mond, B., ‘Generalized convexity and duality in multiple objective programming’, Bull. Austral. Math. Soc. 39 (1989), 287–299.CrossRefGoogle Scholar