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Positivity

, Volume 3, Issue 1, pp 49–64 | Cite as

On Duality in Nonconvex Vector Optimization in Banach Spaces Using Augmented Lagrangians

  • Phan Quoc Khanh
  • Tran Hue Nuong
  • Michel Théra
Article

Abstract

This paper shows how the use of penalty functions in terms of projections on the constraint cones, which are orthogonal in the sense of Birkhoff, permits to establish augmented Lagrangians and to define a dual problem of a given nonconvex vector optimization problem. Then the weak duality always holds. Using the quadratic growth condition together with the inf-stability or a kind of Rockafellar's stability called stability of degree two, we derive strong duality results between the properly efficient solutions of the two problems. A strict converse duality result is proved under an additional convexity assumption, which is shown to be essential.

vector optimization positively proprer minima augmented Lagrangian Birkhoff orthogonality quadratic growth condition inf-stability stability of degree 2 

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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Phan Quoc Khanh
    • 1
  • Tran Hue Nuong
    • 1
  • Michel Théra
    • 2
  1. 1.Département de Mathématiques et d'InformatiqueUniversité d'Hochiminh VilleHochiminh VilleVietnam; E-mail
  2. 2.LACO, UPRESA 6090Université de LimogesLimoges CedexFrance; E-mail

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