Algebraic core and convex calculus without topology

In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the...

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Bibliographic Details
Published in:Optimization Vol. 71; no. 2; pp. 309 - 335
Main Authors: Van Cuong, Dang, Mordukhovich, Boris S., Mau Nam, Nguyen, Cartmell, Addison
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 01.02.2022
Taylor & Francis LLC
Subjects:
Algebra
Algebraic core
coderivatives
Computational geometry
Convex analysis
convex separation
Convexity
Differential calculus
Differential geometry
Mathematical analysis
Mathematical functions
normals
Optimization
subgradients
Theorems
Topology
Vector spaces
ISSN:0233-1934, 1029-4945
Online Access:Get full text
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Summary:In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2020.1800700