Role of Subgradients in Variational Analysis of Polyhedral Functions

Nguyen T. V. Hang (), Woosuk Jung () and Ebrahim Sarabi ()
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Nguyen T. V. Hang: Vietnam Academy of Science and Technology
Woosuk Jung: Miami University
Ebrahim Sarabi: Miami University

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 3, No 10, 1160-1192

Abstract: Abstract 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.

Keywords: Polyhedral functions; Reduction lemma; Nondegenerate solutions; Strict proto-differentiability; Strict twice Epi-differentiability; Proximal mappings; Strong metric regularity; 90C31; 65K99; 49J52; 49J53 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02378-6

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