Role of Subgradients in Variational Analysis of Polyhedral Functions

• Published in: Journal of optimization theory and applications Vol. 200; no. 3; pp. 1160 - 1192
• Main Authors: Hang, Nguyen T. V., Jung, Woosuk, Sarabi, Ebrahim
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Convex analysis; Derivatives; Engineering; Localization; Mathematical functions; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Original Paper; Regularity; Theory of Computation; 90C31; Polyhedral functions; 49J53; Reduction lemma; 49J52; Proximal mappings; Nondegenerate solutions; Strong metric regularity; Strict twice Epi-differentiability; Strict proto-differentiability; 65K99
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-024-02378-6

Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of polyhedral functions, i.e., functions with polyhedral convex epigraphs, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential of polyhedral functions ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. This allows us to characterize continuous differentiability of the proximal mapping and twice continuous differentiability of the Moreau envelope of polyhedral functions. We close the paper with proving the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions.