Properties of efficient solution sets under addition of objectives

This paper is devoted to study the Painlevé–Kuratowski convergence of solution sets for perturbed symmetric set-valued quasi-equilibrium problems (SSQEP)[Formula: see text] via improvement sets. By virtue of the oriented distance function, the sufficient conditions of Painlevé–Kuratowski convergence of efficient solution sets for (SSQEP)[Formula: see text] are obtained through a new nonlinear scalarization technical. Then, under [Formula: see text]-convergence of set-valued mappings, the Painlevé–Kuratowski convergence of weak efficient solution sets for (SSQEP)[Formula: see text] is discussed. What’s more, with suitable convergence assumptions, we also establish the sufficient conditions of lower Painlevé–Kuratowski convergence of Borwein proper efficient solution sets for (SSQEP)[Formula: see text] under improvement sets. Some interesting examples are formulated to illustrate the significance of the main results.

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Stability for the Major Joint Cone Weakly Efficient Solution Sets of Group Multiobjective Programming in Banach Space

2014 Seventh International Joint Conference on Computational Sciences and Optimization ◽

10.1109/cso.2014.56 ◽

2014 ◽

Author(s):

Xuanwei Zhou

Keyword(s):

Banach Space ◽

Efficient Solution ◽

Multiobjective Programming ◽

Weakly Efficient Solution ◽

Solution Sets

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

Mathematics ◽

10.3390/math7100947 ◽

2019 ◽

Vol 7 (10) ◽

pp. 947

Author(s):

Xin Xu ◽

Yang Dong Xu

Keyword(s):

Vector Optimization ◽

Efficient Solution ◽

Optimization Problems ◽

Vector Optimization Problem ◽

Nonlinear Scalarization ◽

Solution Sets ◽

Scalarization Method ◽

Path Connectedness ◽

Weak Efficient Solution ◽

Efficient Solution Set

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.

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