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

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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

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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.",

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AU - Zhang, Yi

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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

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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

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