## Taiwanese Journal of Mathematics

### $\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

#### Abstract

In this paper, we first define a $\mathbb{T}$-cone which is a unified version of several cones, namely, contingent cone, radial cone, $C$-tangent cone, Clarke tangent cone, $S$-cone, adjacent cone, etc. Then, we define the $\mathbb{T}$-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, $S$-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector $\mathbb{T}$-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector $\mathbb{T}$-variational inequality problem are given.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 6 (2010), 2447-2468.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406084

Digital Object Identifier
doi:10.11650/twjm/1500406084

Mathematical Reviews number (MathSciNet)
MR2761608

Zentralblatt MATH identifier
1241.90131

#### Citation

Ansari, Qamrul Hasan; Jahn, Johannes. $\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities. Taiwanese J. Math. 14 (2010), no. 6, 2447--2468. doi:10.11650/twjm/1500406084. https://projecteuclid.org/euclid.twjm/1500406084

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