Further Results on Differential Stability of Convex Optimization Problems

Differential Stability of Convex Discrete Optimal Control Problems

“…The problem of computing the subdifferential and singular subdifferential of µ(·) has been considered in [2] (the Hausdorff locally convex topological vector spaces setting) and in [1] (the Banach space setting). The following result of [2] will be used intensively in this paper.…”

“…In [11], Mordukhovich, Nam and Yen gave formulas for computing and estimating the Fréchet subdifferential, the Mordukhovich subdifferential, and the singular subdifferential of the optimal value function in parametric mathematical programming problems under inclusion constraints. If the problem in question is convex, by using the Moreau-Rockafellar theorem and appropriate regularity conditions, An and Yao [1], An and Yen [2] have obtained formulas for computing subdifferentials of the optimal value function. In some sense, the results of [1] and [2] show that the preceding results of [11] admit a simpler form where several assumptions used in the general nonconvex case can be dropped.…”

“…If the problem in question is convex, by using the Moreau-Rockafellar theorem and appropriate regularity conditions, An and Yao [1], An and Yen [2] have obtained formulas for computing subdifferentials of the optimal value function. In some sense, the results of [1] and [2] show that the preceding results of [11] admit a simpler form where several assumptions used in the general nonconvex case can be dropped.…”

Differential Stability of Convex Optimization Problems with Possibly Empty Solution Sets

“…In [2], An and Yen presented formulas for computing the subdifferential of the optimal value function of convex optimization problems under inclusion constraints in a Hausdorff locally convex topological vector space setting. Afterwards, An and Yao [1] obtained new results on subdifferential of the just mentioned function for problems under geometrical and functional constraints in Banach spaces. In both papers, the authors assumed that the original convex program has a nonempty solution set.…”

“…In both papers, the authors assumed that the original convex program has a nonempty solution set. A natural question arises: Is there any analogous version of the formulas given in [1,2] for the case where the solution set can be empty?…”