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.

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