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Open AccessArticle

Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods

by Xin Xu and Yang Dong Xu *
College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 947; https://doi.org/10.3390/math7100947
Received: 31 August 2019 / Revised: 25 September 2019 / Accepted: 1 October 2019 / Published: 11 October 2019
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
The connectedness and path connectedness of the solution sets to vector optimization problems is an important and interesting study in optimization theories and applications. Most papers involving the direction established the connectedness and connectedness for the solution sets of vector optimization problems or vector equilibrium problems by means of the linear scalarization method rather than the nonlinear scalarization method. The aim of the paper is to deal with the connectedness and the path connectedness for the weak efficient solution set to a vector optimization problem by using the nonlinear scalarization method. Firstly, the union relationship between the weak efficient solution set to the vector optimization problem and the solution sets to a series of parametric scalar minimization problems, is established. Then, some properties of the solution sets of scalar minimization problems are investigated. Finally, by using the union relationship, the connectedness and the path connectedness for the weak efficient solution set of the vector optimization problem are obtained. View Full-Text
Keywords: vector optimization problem; nonlinear scalarization; connectedness; path connectedness vector optimization problem; nonlinear scalarization; connectedness; path connectedness
MDPI and ACS Style

Xu, X.; Xu, Y.D. Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods. Mathematics 2019, 7, 947.

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