Positivity (2005) 9:687–703 © Springer 2005
Optimality Conditions for the Difference
of Convex Set-Valued Mappings
Department of Mathematics, Dhar El Mehrz, Sidi Mohamed Ben Abdellah University, Fes,
Morocco (E-mail: email@example.com)
(Received 18 February 2003; Revised 22 September 2003; Accepted 16 January 2004)
Abstract. Using the concept of subdifferential of cone-convex set valued mappings
recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we
give necessary optimality conditions for nonconvex multiobjective optimization problems.
An example illustrating the usefulness of our results is also given.
Mathematics Subject Classiﬁcation (2000): Primary 90C29, 90C26; Secondary 49K99.
Key words: cone-convex mapping, optimality condition, subdifferential, support function,
In very recent years, the analysis and applications of D.C. mappings (differ-
ence of convex mappings) have been of considerable interest [10, 14, 18,
21]. Genuinely, nonconvex mappings that arise in nonsmooth optimization
are often of this type. Recently, extensive work on the analysis and opti-
mization of D.C. mappings has been carried out [6, 8, 15, 19]. However,
much work remains to be done. For instance, if the data of the objective
function of a standard problem are not exactly known, it makes sense to
replace the objective by a set-valued objective representing fuzzy outcomes.
In this paper, we are concerned with the multiobjective optimization
− min F
subject to x ∈ C and
where X, Y and Z are real normed spaces, C is a nonempty convex closed
subset of X, Y
⊂ Y and Z
⊂ Z are closed convex cones with nonempty
interior, F: X ⇒ Y and G: X ⇒ Y are Y
-convex set valued mappings, and
H: X ⇒ Z and K: X ⇒ Z are Z
-convex set valued mappings.