Subdifferentials of optimal value functions under metric qualification conditions

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Frechet/Mordukhovich/Moreau-Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235-1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369-396, 2009) and of An and Jourani (J Optim Theory Appl 192:82-96, 2022), are also addressed.