scholarly journals Well-Posedness for Tightly Proper Efficiency in Set-Valued Optimization Problem

2011 ◽  
Vol 01 (04) ◽  
pp. 184-186 ◽  
Author(s):  
Yangdong Xu ◽  
Pingping Zhang
Keyword(s):  
Optimization Problem ◽  
Proper Efficiency ◽  
Well Posedness

scholarly journals Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiaohong Hu ◽  
Zhimiao Fang ◽  
Yunxuan Xiong
Keyword(s):  
Saddle Point ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Well Posedness ◽  
Scalar Problem ◽  
New Type ◽  
Strict Efficiency ◽  
Multiplier Theorems

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

scholarly journals Approximate proper efficiency for multiobjective optimization problems

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6091-6101
Author(s):  
Ying Gao ◽  
Zhihui Xu
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problem ◽  
Normal Cone ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Proper Efficiency ◽  
Proximal Normal Cone ◽  
Radiant Set ◽  
Benson Proper Efficiency ◽  
Multiobjective Optimization Problems

This paper is devoted to the study of a new kind of approximate proper efficiency in terms of proximal normal cone and co-radiant set for multiobjective optimization problem. We derive some properties of the new approximate proper efficiency and discuss the relations with the existing approximate concepts, such as approximate efficiency and approximate Benson proper efficiency. At last, we study the linear scalarizations for the new approximate proper efficiency under the generalized convexity assumption and give some examples to illustrate the main results.

scholarly journals Scalarization and well-posedness for set optimization using coradiant sets

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3457-3471
Author(s):  
Bin Yao ◽  
Sheng Li
Keyword(s):  
Optimization Problem ◽  
Set Optimization ◽  
Well Posedness ◽  
Nonlinear Scalarization Function ◽  
Nonlinear Scalarization ◽  
Scalar Optimization ◽  
Set Optimization Problem ◽  
Order Relations ◽  
Scalarization Function

The aim of this paper is to study scalarization and well-posedness for a set-valued optimization problem with order relations induced by a coradiant set. We introduce the notions of the set criterion solution for this problem and obtain some characterizations for these solutions by means of nonlinear scalarization. The scalarization function is a generalization of the scalarization function introduced by Khoshkhabar-amiranloo and Khorram. Moveover, we define the pointwise notions of LP well-posedness, strong DH-well-posedness and strongly B-well-posedness for the set optimization problem and characterize these properties through some scalar optimization problem based on the generalized nonlinear scalarization function respectively.

scholarly journals Well-Posedness by Perturbations for Variational-Hemivariational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Shu Lv ◽  
Yi-bin Xiao ◽  
Zhi-bin Liu ◽  
Xue-song Li
Keyword(s):  
Optimization Problem ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.

scholarly journals Optimizing the Fractional Power in a Model with Stochastic PDE Constraints

2018 ◽  
Vol 18 (4) ◽  
pp. 649-669 ◽  
Author(s):  
Carina Geldhauser ◽  
Enrico Valdinoci
Keyword(s):  
Control Parameter ◽  
Optimization Problem ◽  
Fractional Power ◽  
State Equation ◽  
The State ◽  
Well Posedness ◽  
Stochastic Pde ◽  
Diffusion Operator ◽  
Differentiability Properties ◽  
Fractional Parameter

AbstractWe study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.

scholarly journals A duality theorem for a multiple objective fractional optimization problem

1986 ◽  
Vol 34 (3) ◽  
pp. 415-425 ◽  
Author(s):  
T. Weir
Keyword(s):  
Duality Theorem ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Proper Efficiency ◽  
Dual Solution ◽  
Multiple Objective ◽  
Fractional Optimization ◽  
Linear Problems

Characterizations of efficiency and proper efficiency are given for classes of multiple objective fractional optimization problems. These results are then applied to the case of multiple objective fractional linear problems. A dual problem is given for the multiple objective fractional problem and it is shown that for a properly efficient primal solution the dual solution is also properly efficient.

Hadamard well-posedness for a set-valued optimization problem

2012 ◽  
Vol 7 (3) ◽  
pp. 559-573 ◽  
Author(s):  
J. Zeng ◽  
S. J. Li ◽  
W. Y. Zhang ◽  
X. W. Xue
Keyword(s):  
Optimization Problem ◽  
Well Posedness

Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1168-1182
Author(s):  
Hongwei Liang ◽  
Zhongping Wan
Keyword(s):  
Optimization Problem ◽  
Saddle Points ◽  
Sufficient Conditions ◽  
Proper Efficiency ◽  
Solid Cone ◽  
Linear Spaces ◽  
New Class ◽  
Benson Proper Efficiency ◽  
Real Linear ◽  
Necessary And Sufficient

Abstract A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

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