Positivity 13 (2009), 657–669
2009 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040657-13, published online February 6, 2009
Necessary optimality conditions for
set-valued optimization problems via the
Nazih Gadhi and Lahoussine Lafhim
Abstract. The paper concerns ﬁrst-order necessary optimality conditions for
set-valued optimization problems. Based on the extremal principle developed
by Mordukhovich , we derive fuzzy/approximate necessary optimality con-
ditions. An example that illustrates the usefulness of our results is given.
Mathematics Subject Classiﬁcation (2000). Primary 90C29; Secondary 49K99.
Keywords. Asplund space, Extremal principle, Fr´echet normal cone,
Mordukhovich normal cone, Set-valued mapping.
Nowadays set-valued optimization means set-valued analysis and its application
to optimization, and it is an extension of continuous optimization to set-valued
case. In this research area one investigates optimization problems with constraints
and/ or an objective function described by set-valued maps, or investigations in
set-valued analysis are applied to standard optimization problems. In the last
decade there has been an increasing interest in set-valued optimization.
General optimization problems with set-valued constraints or a set-valued
objective function are closely related to problems in stochastic programming, fuzzy
programming and optimal control. If the values of a given function vary in a
speciﬁed region, this fact could be described by using a membership function in
the theory of fuzzy sets or using information on the distribution of the func-
tion values. In this general setting probability distributions or membership func-
tions are not needed because only sets are considered. Optimal control problems
with differential inclusions belong to this class of set-valued optimization problems
as well. Set-valued optimization seems to have the potential to become a bridge
between different areas in optimization. And it is a substantial extension of stan-
dard optimization theory. Set-valued analysis is the most important tool for such