Step-affine functions, halfspaces, and separation of convex sets with applications to convex optimization problems
type: Item
instance of: scholarly article
title: Step-affine functions, halfspaces, and separation of convex sets with applications to convex optimization problems
zbMATH Open document ID: 1482.90150
DOI: 10.1134/S008154382103010X
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2021-08-03T00:00:00Z
review text: The paper presents analytical criteria for the disjointness of convex subsets of a real vector space. The criteria are based on the separation of convex sets by step-linear and step-affine functions. These criteria are essentially analytic versions of the Kakutani-Tukey criterion on the separation of convex sets by halfspaces and these are important due to their applications to irregular convex optimization problems. As applications of these results, the author derives a minimality criterion for solutions of convex vector optimization problems considered in real vector spaces without topology and an optimality criterion for admissible points in classical convex programming problems not satisfying the Slater regularity condition. Some open questions related to the separation of convex sets by halfspaces are also given.
reviewed by: Samir Kumar Neogy
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C29
zbMATH DE Number: 7377484
zbMATH Keywords: step-affine functions
zbMATH Keywords: halfspaces
zbMATH Keywords: separation of convex sets
zbMATH Keywords: convex vector optimization problems
zbMATH Keywords: convex programming
author: V. V. Gorokhovik
MaRDI profile type: Publication
cites work: Ein Beweis des Satzes von M. Eidelheit über konvexe Mengen
cites work: Q5784671
cites work: Q3244654
cites work: The structure of hemispaces in \({\mathbb{R}}^ n\)
cites work: Q3937918
cites work: Q4002048
cites work: Q4821265
cites work: Classification of semispaces according to their types in infinite-dimensional vector spaces
cites work: Q4220907
cites work: Q4525504
cites work: Lexicographical representation of convex sets
cites work: On constructing total orders and solving vector optimization problems with total orders
cites work: The generalization of total ordering cones and vectorization to separable Hilbert spaces
cites work: The structure of semispaces
cites work: Maximal Separation Theorems for Convex Sets
cites work: Q3884743
cites work: Q4714503
cites work: Maximal convex sets
cites work: Convex half-spaces
cites work: Q3685582
cites work: Lexicographical separation in \({\mathbb{R}}^ n\)
cites work: Q3966610
full work available at URL: s008154382103010x
OpenAlex ID: W3184640751
description: scientific article; zbMATH DE number 7377484