  The directional subdifferential of the value function gives an estimate on
how much the optimal value changes under a perturbation in a certain direction.
In this paper we derive upper estimates for the directional limiting and
singular subdifferential of the value function for a very general parametric
optimization problem. We obtain a characterization for the directional
Lipschitzness of a locally lower semicontinuous function in terms of the
directional subdifferentials. Based on this characterization and the derived
upper estimate for the directional singular subdifferential, we are able to
obtain a sufficient condition for the directional Lipschitzness of the value
function. Finally, we specify these results for various cases when all
functions involved are smooth, when the perturbation is additive, when the
constraint is independent of the parameter, or when the constraints are
equalities and inequalities. Our results extend the corresponding results on
the sensitivity of the value function to allow directional perturbations. Even
in the case of full perturbations, our results recover or even extend some
existing results, including the Danskin's theorem.
