Painlevé-Kuratowski convergences of the solution sets for vector optimization problems with free disposal sets

Nguyen Minh Tung; Mai Van Duy

doi:10.3934/jimo.2021066

Article Contents

2022, Volume 18, Issue 4: 2255-2276. Doi: 10.3934/jimo.2021066

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$ k $

-means problem with penalty

Painlevé-Kuratowski convergences of the solution sets for vector optimization problems with free disposal sets

Nguyen Minh Tung^1, , and
Mai Van Duy^2,

1.
Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
2.
Department of Mathematics and Computing, University of Science, Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam

^* Corresponding author: Nguyen Minh Tung
^* Corresponding author: Nguyen Minh Tung

Dedicated to Professor Phan Quoc Khanh on the occasion of his 75th birthday

Received: April 2020

Revised: January 2021

Published: July 2022

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Abstract

This paper aims to present results on the Painlev$ \mathrm{\acute{e}} $-Kuratowski set-convergence of the sets of both infimal and minimal points of a sequence of perturbed vector optimization problems through free disposal sets. By assumptions of sequential compactness of the feasible sets or the uniform coerciveness of objective functions, this convergence is obtained both in the image and given spaces under the perturbations of objective functions and feasible sets. Besides, we also establish set-convergences of sequences of approximation solution sets. Applications to the stability of conic, quadratic and linear vector optimization problems are given. Some examples are provided to illustrate our results.

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Mathematics Subject Classification: 49K40, 90C26, 90C31.

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Figure 1. The free disposal set $ E $ and some infimal sets

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Figure 2. The free disposal set $ E $ and $ E $-infimal sets in Example 6

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Figure 3. The free disposal set $ E $ and $ E $-infimal sets in Example 7

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