Differential Stability in Convex Optimization via Generalized Polyhedrality

The problem of computing/estimating the subdifferential of the optimal value function of a parametric optimization problem described by a proper generalized polyhedral convex function and a generalized polyhedral convex set-valued map is considered in full extent for the first time in this paper. The Hausdorff locally convex topological vector spaces setting is adopted. Upper estimates and lower estimates for the subdifferential, as well as for the singular subdifferential, of the optimal value function at a given parameter are established under just one assumption: the solution set of the original optimization problem is nonempty. A carefully designed example is provided to show that all the obtained estimates are sharp in the sense that they can be attained. Some open questions are given.