On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from Hogan (SIAM Rev 15:591–603, 1973a) for abstract feasible set mappings under complete convexity as well as standard differentiability results from Hogan (Oper Res 21:188–209, 1973b) for feasible set mappings in functional form under the Slater condition in the unfolded feasible set. In particular, we present sufficient conditions for the inner semi‐continuity of feasible set mappings and, using techniques from nonsmooth analysis, provide functional descriptions of tangent cones to the domain of the optimal value function. The latter makes the stated directional differentiability results accessible for practical applications.