On Differential Stability of a Class of Convex Optimization Problems

The recent results of An, Luan, and Yen [Differential stability in convex optimization via generalized polyhedrality. Vietnam Journal of Mathematics 53 , 721–734 (2025)] on differential stability of parametric optimization problems described by proper generalized polyhedral convex functions and gene...

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Published in:Journal of optimization theory and applications Vol. 208; no. 1; p. 41
Main Authors: Yen, Nguyen Dong, An, Duong Thi Viet, Huong, Vu Thi, Luan, Nguyen Ngoc
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2026
Springer Nature B.V
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Convex analysis
Convexity
Engineering
Estimates
Investigations
Linear programming
Mathematical functions
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Stability
Theorems
Theory of Computation
Vector spaces
90C31
Parametric optimization problem in Hausdorff locally convex topological vector spaces
Generalized polyhedral convex set-valued map
Sum rule
90C30
49K40
90C25
Subdifferential
Singular subdifferential
The optimal value function
Generalized polyhedral convex function
49J27
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:The recent results of An, Luan, and Yen [Differential stability in convex optimization via generalized polyhedrality. Vietnam Journal of Mathematics 53 , 721–734 (2025)] 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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-025-02856-5