Ok, before anyone responds with the fact that the optimal value function of an LP is not twice differentiable.....I know, I know. I am not sure how to write what I am asking which is a big part of why I am asking the question. I am trying to prove that the marginal value of the LP (a two stage stochastic LP to be specific), call it $\Pi$, with respect to one parameter, lets say $p_1$ weakly decreases in another parameter $p_2$. Using Gal and Greenberg "Advances in Sensitivity Analysis and Parametric Programming" I can find the directional derivatives in terms of the min and max over the dual variables. Then I have an expression for the dual in terms of parameters and primal variables. I want to take a derivative of this since I can bound the change in these variables in nice ways but I don't know how to take this derivative. Intuitively, something like a distributional derivative seems the most appropriate but I can't find any guides that show how to do this rigorously. On the other hand I have looked at Rockafellar and Wets "Variational Analysis" and they use a second epiderivative. I feel like I have a decent understanding of the Gal and Greenberg directional derivatives and am hesitant to switch over to the Rockafellar and Wets approach. So I wanted to ask others' opinion and thoughts on the topic. Also, are the first directional derivatives and the epiderivatives the same idea?
