Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications

2014 ◽  
Vol 24 (1) ◽  
pp. 363-384 ◽  
Author(s):  
Xi Yin Zheng ◽  
Kung Fu Ng
Keyword(s):  
Banach Spaces ◽  
Normal Cone ◽  
Proximal Normal Cone

scholarly journals Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Spaces ◽  
Normal Cone ◽  
Density Theorem ◽  
Space Applications ◽  
Proximal Subdifferential ◽  
Proximal Normal Cone ◽  
Uniformly Smooth ◽  
Calculus Rules ◽  
New Concepts ◽  
Uniformly Smooth Banach Spaces

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex andq-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.

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 New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

2021 ◽  
Author(s):  
A. Hantoute ◽  
M. A. López-Cerdá
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Normal Cone ◽  
Countable Family ◽  
Weighted Sums ◽  
Reflexive Banach Spaces ◽  
Finite Dimensional ◽  
Finite Sums ◽  
Effective Domain ◽  
The Relationship

AbstractThis paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family.

scholarly journals Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Si-Huan Li ◽  
Qiang Wang ◽  
Shu Xu ◽  
Jun-Xiang Wang
Keyword(s):  
Optimality Conditions ◽  
Banach Spaces ◽  
Normal Cone ◽  
Equilibrium Problems ◽  
Necessary Optimality Conditions ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Vector Equilibrium Problems ◽  
Vector Equilibrium

The concept of sharp efficient solution for vector equilibrium problems on Banach spaces is proposed. Moreover, the Fermat rules for local efficient solutions of vector equilibrium problems are extended to the sharp efficient solutions by means of the Clarke generalized differentiation and the normal cone. As applications, some necessary optimality conditions and sufficient optimality conditions for local sharp efficient solutions of a vector optimization problem with an abstract constraint and a vector variational inequality are obtained, respectively.

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