Positivity 3: 49–64, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
On Duality in Nonconvex Vector Optimization in
Banach Spaces Using Augmented Lagrangians
PHAN QUOC KHANH
, TRAN HUE NUONG
and MICHEL THÉRA
Département de Mathématiques et d’Informatique, Université d’Hochiminh Ville, 227 rue Nguyen
Van Cu, Q.1, Hochiminh Ville – Vietnam; E-mail: email@example.com;
6090, Université de Limoges, 87060 Limoges Cedex, France; E-mail: firstname.lastname@example.org
(Received: 18 February 1998; Accepted: 20 July 1998)
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.
Mathematics Subject Classiﬁcation (1991): 90C29
Key words: vector optimization, positively proprer minima, augmented Lagrangian, Birkhoff or-
thogonality, quadratic growth condition, inf-stability, stability of degree 2
Let X be a set, Y be a topological vector space and Z be a Banach space, Y and Z
being ordered by closed convex cones K and M respectively. Let F : X → Y and
G : X → Z be two mappings.
Consider the vector optimization problem
It is well known that whenever Y =
and X is a linear space, if problem (P ) is
convex, i.e. F is convex and G is M-convex, one obtains the Lagrangian duality
between (P ) and its Lagrange dual. If (P ) is nonconvex, a nonzero duality gap
appears. To derive duality results for nonconvex problems there are three major
ways to overcome this duality gap.
The research reported here was sponsored in part by the FICU-AUPELF program “Coop´eration
en math´ematiques appliqu´ees entre la Belgique, la France et le Viêtnam”. The research of the ﬁrst
two authors was also supported by the the Vietnam National Basic Research Program in Natural