A Multiplier Rule for Stable Problems in Vector Optimization

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.