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Date Published: 15/06/2015
Online Published: 15/06/2015
Abstract Views: 67
Views PDF: 36
DOI: https://doi.org/10.52714/dthu.13.6.2015.206
Issue
No. 13 (2015): Part B - Natural Sciences
Section
Natural Sciences Issue
How to Cite
Vo, D. T. (2015). The directionally normal cone and optimal condition. Dong Thap University Journal of Science, 13, 63-68. https://doi.org/10.52714/dthu.13.6.2015.206

The directionally normal cone and optimal condition

Duc Thinh Vo1,
1 Dong Thap University

Main Article Content

Abstract

This paper is devoted to study some properties of the directionally Fréchet normal cones and the directionally limiting normal cones. Moreover, we also modify those directionally limiting normal cones and establish some properties of the modified normal cones. Then we provide some examples to illustrate the differences between these normal cones. Finally, we generate the concepts of the directionally sub-differential via the directionally normal cones. By using the directionally sub-differentials, we come up with a necessary condition for the directionally optimal solution of an optimization problem.

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Keywords

Directionally normal cone, directionally sub-differential, optimality condition.

References

[1]. N. L. H. Anh and P. Q. Khanh (2013), “Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives”, J. Glob. Optim., (56), p. 519-536.
[2]. I. Ginchev and B. S. Mordukhovich (2011), “On directionally dependent subdifferentials”, C. R. Acad. Bulgare Sci., (64), p. 497-508.
[3]. I. Ginchev and B. S. Mordukhovich (2012), “Directional subdifferentials and optimality conditions”, Positivity, (16), p. 707-737.
[4]. P. Q. Khanh and L. T. Tung (2013), “First and second-order optimality conditions using approximations for vector equilibrium problems with constraints”, J. Glob. Optim., (55), p. 901-920.
[5]. B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation I, Springer, Berlin.
[6]. J. P. Penot (2014), “Directionally limiting subdifferentials and second-order optimality conditions”, Optim. Lett., (8), p. 1191-1200.

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