The notion of ˜ℓ-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of ˜ℓ-stable functions coincides with the class of C1,1 functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.
Vsevolod I. Ivanov stated (Nonlinear Analysis 125 (2015), 270-289) the general second-order optimality condition for the constrained vector problem in terms of Hadamard derivatives. We will consider its special case for a scalar problem and show some corollaries for example for ℓ-stable at feasible point functions. Then we show the advantages of obtained results with respect to the previously obtained results.