Sensitivity Analysis of Optimal Value Functions of Convex Parametric Programs with Possibly Empty Solution Sets

Let $g( y_1 ,y_2 )$ be the optimal value of the abstract program which consists in minimizing a convex function $f:X \to \mathbb{R} \cup \{ + \infty \}$ over a feasible set of the form $\{ x \in X : A_1 x = y_1 ,A_2 x \leq_K y_2 \}$. Without assuming the existence of optimal solutions to this minimization problem, we derive formulas for the subdifferential and the approximate subdifferential of g. Several applications are discussed.