The directionally normal cone and optimal condition
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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
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Directionally normal cone, directionally sub-differential, optimality condition.
References
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[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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