As a complement to a recent paper by An and Yen (Appl Anal 94:108–
128, 2015) on subdifferentials of the optimal value function in parametric convex
programming under inclusion constraints and functional constraints, this paper studies
the differential stability of convex optimization problems under a regularity condition
of Aubin’s type (Aubin in Optima and equilibria: an introduction to nonlinear analysis.
Springer, New York, 1998). By a suitable sum rule for convex subdifferentials, we
obtain exact formulas for the subdifferential and singular subdifferential of the optimal
value function. Illustrative examples and a detailed comparison of our results with
those of the above-mentioned paper are given.