On the directional derivative of the optimal solution mapping without linear independence constraint qualification

In the paper it is shown that if a strong second-order sufficient condition and Slater’s condition hold at a minimizer of a convex programming problem then, for sufficiently smooth perturbations of the problem functions the optimal solution map admits a directional derivative in every direction. This directional derivative may be computed solving certain system of linear equations and inequalities. Furthermore, a special element of the upper Dini derivative of the Karush-Kuhn-Tucker set mapping is computed.