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 Coderivatives and Aubin property of efficient point and efficient solution set-valued maps in parametric vector optimization

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaowei Xue
Keyword(s):  
Explicit Expression ◽  
Vector Optimization ◽  
Efficient Solution ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Efficient Point ◽  
Aubin Property ◽  
Parametric Vector Optimization ◽  
Whole Space ◽  
Efficient Solution Set

Abstract The aim of this paper is computing the coderivatives of efficient point and efficient solution set-valued maps in a parametric vector optimization problem. By using a method different from the existing literature we establish an upper estimate and explicit expression for the coderivatives of an efficient point set-valued map where the independent variable can take values in the whole space. As an application, we give some characterizations on the Aubin property of an efficient point map and an explicit expression of the coderivative for an efficient solution map. We provide several examples illustrating the main results.

Efficiency and duality for vector optimization problem on Riemannian manifolds involving KT-B-invexity

2019 ◽  
Vol 12 (07) ◽  
pp. 1950088
Author(s):  
Babli Kumari ◽  
Anurag Jayswal
Keyword(s):  
Vector Optimization ◽  
Riemannian Manifolds ◽  
Efficient Solution ◽  
Optimization Problem ◽  
Dual Problem ◽  
Vector Optimization Problem ◽  
Converse Duality ◽  
Duality Results ◽  
Weak Efficient Solution

In this paper, we consider a vector optimization problem on Riemannian manifolds for which we define KT-B-invex and KT-B-pseudoinvex functions. Further, we prove that every vector Kuhn–Tucker point is a weak efficient solution for considered vector optimization problem under the suitable assumptions. Moreover, we also study the Mond–Weir dual problem for the aforesaid problem and establish its weak, strong and converse duality results.

scholarly journals A note of stability of weakly efficient solution set for optimization with set-valued maps

2003 ◽  
Vol 16 (3) ◽  
pp. 267-273
Author(s):  
Luo Qun
Keyword(s):  
Efficient Solution ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Baire Category ◽  
Efficient Solutions ◽  
Solution Set ◽  
Weakly Efficient Solution ◽  
Solution Sets ◽  
Essential Components ◽  
Efficient Solution Set

In this paper, we study the stability of weakly efficient solution sets for optimization problems with set-valued maps. We introduce the concept of essential weakly efficient solutions and essential components of weakly efficient solution sets. We first show that most optimization problems with set-valued maps (in the sense of Baire category) are stable. Secondly, we obtain some sufficient conditions for the existence of one essential weakly efficient solution or one essential component of the weakly efficient solution set .

scholarly journals Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Qinghai He ◽  
Weili Kong
Keyword(s):  
Banach Spaces ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Solution Set ◽  
Pareto Optimal ◽  
Pareto Solution ◽  
Value Set ◽  
Set Valued Mapping ◽  
Optimal Value

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

scholarly journals Mixed E-duality for E-differentiable Vector Optimization Problems Under (Generalized) V-E-invexity

2021 ◽  
Vol 2 (3) ◽  
Author(s):  
Najeeb Abdulaleem
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Vector Optimization Problem ◽  
Equality Constraints ◽  
Vector Optimization Problems ◽  
Duality Theorems ◽  
Differentiable Vector

AbstractIn this paper, a class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. The so-called vector mixed E-dual problem is defined for the considered E-differentiable vector optimization problem with both inequality and equality constraints. Then, several mixed E-duality theorems are established under (generalized) V-E-invexity hypotheses.

scholarly journals Lipschitz and differentiable dependence of solutions on a parameter in a scalarization method

1987 ◽  
Vol 42 (3) ◽  
pp. 353-364 ◽  
Author(s):  
Alicia Sterna-Karwat
Keyword(s):  
Vector Optimization ◽  
Normed Space ◽  
Convex Cone ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Optimal Solutions ◽  
Scalar Problem ◽  
Lipschitz Continuous ◽  
Scalarization Method ◽  
Convexity Assumptions

AbstractThis paper is concerned with a vector optimization problem set in a normed space where optimality is defined through a convex cone. The vector problem can be solved using a parametrized scalar problem. Under some convexity assumptions, it is shown that dependence of optimal solutions on the parameter is Lipschitz continuous. Hence differentiable dependence on the solutions on the parameter is derived.

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