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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Vydáno v:	Journal of optimization theory and applications Ročník 208; číslo 1; s. 41
Hlavní autoři:	Yen, Nguyen Dong, An, Duong Thi Viet, Huong, Vu Thi, Luan, Nguyen Ngoc
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.01.2026 Springer Nature B.V
Témata:	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
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Abstract	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.
AbstractList	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. 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.
ArticleNumber	41
Author	Luan, Nguyen Ngoc Yen, Nguyen Dong Huong, Vu Thi An, Duong Thi Viet
Author_xml	– sequence: 1 givenname: Nguyen Dong surname: Yen fullname: Yen, Nguyen Dong email: ndyen@math.ac.vn organization: Institute of Mathematics, Vietnam Academy of Science and Technology – sequence: 2 givenname: Duong Thi Viet orcidid: 0000-0003-1822-4929 surname: An fullname: An, Duong Thi Viet organization: Department of Mathematics and Informatics, Thai Nguyen University of Sciences – sequence: 3 givenname: Vu Thi orcidid: 0009-0007-4869-0505 surname: Huong fullname: Huong, Vu Thi organization: Institute of Mathematics, Vietnam Academy of Science and Technology – sequence: 4 givenname: Nguyen Ngoc orcidid: 0000-0002-9645-7923 surname: Luan fullname: Luan, Nguyen Ngoc organization: Department of Mathematics and Informatics, Hanoi National University of Education
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Cites_doi	10.1080/02331934.2020.1836637 10.1080/00036811.2014.890710 10.1007/s10957-021-01959-z 10.1007/978-3-662-02959-6_1 10.1515/9781400873173 10.1007/s10898-019-00763-4 10.1007/s10957-018-1431-1 10.1007/s11228-021-00601-4 10.1007/978-1-4419-9467-7 10.1007/978-1-4612-1394-9 10.1007/978-3-031-26458-0 10.1007/s11228-017-0426-7 10.1080/02331934.2024.2420679 10.1080/02331934.2019.1687697 10.1007/s10013-024-00721-y 10.1007/978-3-030-94785-9 10.1080/01630563.2017.1387863
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References	NN Luan (2856_CR9) 2022; 193 DTV An (2856_CR3) 2025; 53 2856_CR6 2856_CR14 NN Luan (2856_CR10) 2025; 74 DTV An (2856_CR4) 2019; 181 NN Luan (2856_CR11) 2019; 75 JF Bonnans (2856_CR8) 2000 BS Mordukhovich (2856_CR13) 2023 DTV An (2856_CR1) 2021; 29 NN Luan (2856_CR12) 2018; 39 HH Bauschke (2856_CR7) 2011 RT Rockafellar (2856_CR16) 1970 NT Toan (2856_CR18) 2022; 71 BS Mordukhovich (2856_CR15) 2017; 25 DTV An (2856_CR2) 2020; 69 W Rudin (2856_CR17) 1991 DTV An (2856_CR5) 2015; 94
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SubjectTerms	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
Title	On Differential Stability of a Class of Convex Optimization Problems
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