On scalar and vector ℓℓ-stable functions

The notion of a scalar function that is ℓℓ-stable at a point is introduced in [D. Bednařík, K. Pastor, On second-order conditions in unconstrained optimization, Math. Program. Ser. A 113 (2008) 283–298]. In the present paper a characterization of the ℓℓ-stable functions is obtained. Further, the notion of an ℓℓ-stable function is generalized from scalar to vector functions. In an application, optimality conditions for constrained vector problems with ℓℓ-stable data are established.