Journal of Convex Analysis 19 (2012), No. 2, 525--539
Copyright Heldermann Verlag 2012
A Multiplier Rule for Stable Problems in Vector Optimization
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
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.
Keywords: Multiplier rules, stable maps, set-valued analysis, contingent epiderivatives, contingent cones, weak-minimizers.
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