Generalized Lagrange multipliers:
regularity and boundedness
Giancarlo Bigi and Massimo Pappalardo
Necessary optimality conditions for vector optimization problems can
be established exploiting generalized derivatives. In this paper we
introduce a generalized Dini derivative for vector-valued functions
via Kuratowski limit, with a global operator and not with a
componentwise one; moreover, we employ such derivative to obtain
optimality conditions, which are formulated in terms
of the impossibility of positively homogeneous systems. By
linear separation arguments and adding convexlikeness assumptions on
this vector Dini derivative, we obtain a set of generalized Lagrange
multipliers for vector minimum points; such multipliers can be viewed
as the gradient of the separating hyperplane.
The subsequent analysis is devoted to establish regularity conditions
(or constraint qualification) in order to have additional properties
on these multipliers. Most of this analysis has been developed in
recent papers. Here, we apply the result of those papers
to the multipliers obtained with this vector Dini derivative. The
main result of this paper is to show that the introduced regularity
conditions are equivalent to the boundedness of the set of multipliers
and generalize classical results of the scalar case.
If you are interested in this paper, feel free to contact me.
My Home Page