On Differential Stability of a Class of Convex Optimization Problems

TL;DR

This paper refines and extends existing results on the differential stability of convex optimization problems, providing exact formulas for subdifferentials and clarifying conditions for their estimates in a generalized polyhedral setting.

Contribution

It develops sharper relationships between subdifferential estimates and provides exact formulas, especially under polyhedral convexity assumptions.

Findings

Upper estimates for subdifferentials are always exact.

Lower estimates can be strict, showing a gap with upper estimates.

Exact formulas are obtained for subdifferentials and singular subdifferentials.

Abstract

The recent results of An, Luan, and Yen [Differential stability in convex optimization via generalized polyhedrality. Vietnam J. Math. https://-doi.org/10.1007/s10013-024-00721-y] on differential stability of parametric optimization problems described by proper generalized polyhedral convex functions and generalized polyhedral convex set-valued maps are analyzed, developed, and sharpened in this paper. Namely, keeping the Hausdorff locally convex topological vector spaces setting, we clarify the relationships between the upper estimates and lower estimates for the subdifferential and the singular subdifferential of the optimal value function. As shown by an example, the lower estimates can be strict. But, surprisingly, each upper estimate is an equality. Thus, exact formulas for the subdifferential and the singular subdifferential under consideration are obtained. In addition, it is proved that each subdifferential upper estimate coincides with the corresponding lower estimate if either the objective function or the constraint set-valued map is polyhedral convex.