Abstract

Recently, by using the derivatives of scalarized maps, associated with a vector optimization problem, new multiplier rules have been proven. The first objective of this paper is to show that those rules do not hold in infinite dimensional setting without imposing additional restrictions, even when the ordering cone has a nonempty interior. In this paper, we employ the weak-interior of the ordering cone to propose a new condition. Under this condition, we show that the original problem is equivalent to an scalarized finite-dimensional problem. As a consequence we prove a multiplier rule in infinite dimensional setting for stable data. The proof of these results rely on a new estimate about the dual cones of weakly-solid cones. Several counterexamples showing that the hypotheses are essential are given.

Author information

Contact details are reproduced from the original publication and may be historical.

Akhtar A. Khan

Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, U.S.A.

aaksma@rit.edu

Miguel Sama

Dep. de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040 Madrid, Spain

msama@ind.uned.es

Keywords

  • Multiplier rules
  • stable maps
  • set-valued analysis
  • contingent epiderivatives
  • contingent cones
  • weak-minimizers

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Suggested citation

A. A. Khan, M. Sama. “A Multiplier Rule for Stable Problems in Vector Optimization.” Journal of Convex Analysis 19 (2012), No. 2, 525–539. https://doi.org/10.68381/jca19029

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.