  This paper is concerned with the directional derivative of the value function
for a very general set-constrained optimization problem under perturbation.
Under reasonable assumptions, we obtain upper and lower estimates for the upper
and lower Dini directional derivative of the value function respectively, from
which we obtain Hadamard directional differentiability of the value function
when the set of multipliers is a singleton. Our results do not require
convexity of the set involved. Even in the case of a parametric nonlinear
program, our results improve the classical ones in that our regularity
conditions are weaker and the directional solution set is used which is in
general smaller than its nondirectional counterparts.
