A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

KEYWORDS:

• Lagrange multiplier
• convex vector optimization
• process
• optimality conditions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C29
• 90C25
• 90C31
• 90C48

Acknowledgments

We thank the referee for him/her suggestions which have helped us to improve the overall aspect of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Fernando García-Castaño http://orcid.org/0000-0002-8352-8235

Additional information

Funding

The authors have been supported by project MTM2017-86182-P (AEI, Spain and ERDF/FEDER, EU). The author Fernando García-Castaño has also been supported by MINECO and FEDER (MTM2014-54182).