Convergence Results for Henig Proper Efficient Solution Sets of Vector Optimization Problems

2014 ◽  
Vol 35 (11) ◽  
pp. 1419-1434 ◽  
Author(s):  
Xiao-bing Li ◽  
Zai-yun Peng ◽  
Zhi Lin
Keyword(s):  
Vector Optimization ◽  
Efficient Solution ◽  
Optimization Problems ◽  
Vector Optimization Problems ◽  
Solution Sets ◽  
Convergence Results ◽  
Proper Efficient Solution

On Second-Order Composed Proto-Differentiability of Proper Perturbation Maps in Parametric Vector Optimization Problems

2020 ◽  
pp. 2050040
Author(s):  
Le Thanh Tung
Keyword(s):  
Vector Optimization ◽  
Efficient Solution ◽  
Optimization Problems ◽  
Second Order ◽  
Vector Optimization Problems ◽  
Solution Maps ◽  
Qualification Conditions ◽  
Parametric Vector Optimization ◽  
Perturbation Maps ◽  
Proper Efficient Solution

The main aim of this paper is to study second-order sensitivity analysis in parametric vector optimization problems. We prove that the proper perturbation maps and the proper efficient solution maps of parametric vector optimization problems are second-order composed proto-differentiable under some appropriate qualification conditions. Some examples are provided to illustrate our results.

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

Mathematics ◽  
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.

scholarly journals Set optimization using improvement sets

2017 ◽  
Vol 27 (2) ◽  
pp. 153-167 ◽  
Author(s):  
M. Dhingra ◽  
C.S. Lalitha
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Existence Theorems ◽  
Minimal Solution ◽  
Set Optimization ◽  
Vector Optimization Problems ◽  
Optimal Solutions ◽  
Solution Sets ◽  
Improvement Sets

In this paper we introduce a notion of minimal solutions for set-valued optimization problem in terms of improvement sets, by unifying a solution notion, introduced by Kuroiwa [15] for set-valued problems, and a notion of optimal solutions in terms of improvement sets, introduced by Chicco et al. [4] for vector optimization problems. We provide existence theorems for these solutions, and establish lower convergence of the minimal solution sets in the sense of Painlev?-Kuratowski.

∊-STRICTLY EFFICIENT SOLUTIONS OF VECTOR OPTIMIZATION PROBLEMS WITH SET-VALUED MAPS

2007 ◽  
Vol 24 (06) ◽  
pp. 841-854 ◽  
Author(s):  
TAIYONG LI ◽  
YIHONG XU ◽  
CHUANXI ZHU
Keyword(s):  
Saddle Point ◽  
Vector Optimization ◽  
Efficient Solution ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Lagrangian Multiplier ◽  
Multiplier Theorem ◽  
Lagrangian Multiplier Theorem

In this paper, the notion of ∊-strictly efficient solution for vector optimization with set-valued maps is introduced. Under the assumption of the ic-cone-convexlikeness for set-valued maps, the scalarization theorem, ∊-Lagrangian multiplier theorem, ∊-saddle point theorems and ∊-duality assertions are established for ∊-strictly efficient solution.

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