On the structure of optimal solutions in a mathematical programming problem in a convex space

Alexey Piunovskiy; Yi Zhang

doi:10.1016/j.orl.2023.07.006

On the structure of optimal solutions in a mathematical programming problem in a convex space

Alexey Piunovskiy, Yi Zhang^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

17 Downloads (Pure)

Abstract

We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.

Original language	English
Pages (from-to)	488-493
Number of pages	6
Journal	Operations Research Letters
Volume	51
Issue number	5
Early online date	28 Jul 2023
DOIs	https://doi.org/10.1016/j.orl.2023.07.006
Publication status	Published - Sept 2023

Bibliographical note

Funding Information:
We thank the anonymous referee for the careful reading and helpful remarks. In particular, Remark 2.1 is suggested by him/her.

Publisher Copyright:
© 2023 The Author(s)

Keywords

Extreme point
Feinberg-Shwartz lemma
Mixed optimal solution
Problem with constraints

ASJC Scopus subject areas

Software
Management Science and Operations Research
Industrial and Manufacturing Engineering
Applied Mathematics

Access to Document

10.1016/j.orl.2023.07.006Licence: Creative Commons: Attribution (CC BY)

PiunovskiyA2023structureFinal published version, 329 KBLicence: Creative Commons: Attribution (CC BY)

Cite this

@article{b25c2beeb278461089e5b215c6444e3f,

title = "On the structure of optimal solutions in a mathematical programming problem in a convex space",

abstract = "We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.",

keywords = "Extreme point, Feinberg-Shwartz lemma, Mixed optimal solution, Problem with constraints",

author = "Alexey Piunovskiy and Yi Zhang",

note = "Funding Information: We thank the anonymous referee for the careful reading and helpful remarks. In particular, Remark 2.1 is suggested by him/her. Publisher Copyright: {\textcopyright} 2023 The Author(s)",

year = "2023",

month = sep,

doi = "10.1016/j.orl.2023.07.006",

language = "English",

volume = "51",

pages = "488--493",

journal = "Operations Research Letters",

issn = "0167-6377",

publisher = "Elsevier",

number = "5",

}

TY - JOUR

T1 - On the structure of optimal solutions in a mathematical programming problem in a convex space

AU - Piunovskiy, Alexey

AU - Zhang, Yi

PY - 2023/9

Y1 - 2023/9

N2 - We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.

AB - We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems.

KW - Extreme point

KW - Feinberg-Shwartz lemma

KW - Mixed optimal solution

KW - Problem with constraints

UR - http://www.scopus.com/inward/record.url?scp=85166471721&partnerID=8YFLogxK

U2 - 10.1016/j.orl.2023.07.006

DO - 10.1016/j.orl.2023.07.006

M3 - Article

AN - SCOPUS:85166471721

SN - 0167-6377

VL - 51

SP - 488

EP - 493

JO - Operations Research Letters

JF - Operations Research Letters

IS - 5

ER -

On the structure of optimal solutions in a mathematical programming problem in a convex space

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Fingerprint

Cite this