NTMSCI 6, No. 1, 166-172 (2018) 166 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2018.259 Generalized composed radial epiderivatives Gonca Inceoglu Department of Mathematics and Science Education, Anadolu University, Turkey Received: 24 November 2017, Accepted: 29 January 2018 Published online: 7 March 2018. Abstract: In this paper, the generalized composed radial epiderivative for set-valued maps is introduced and some of its properties are investigated. Existence conditions for the generalized composed radial epiderivative are established. Keywords: Radial cone, generalized composed radial epiderivative, set-valued optimization. 1 Introduction In the last thirty years the notion of derivatives or epiderivatives has been formulated in different ways. (see [1, 2, 7, 8, 10, 13, 19, 23, 24, 25, 28, 31]). Aubin first introduced the notion of the contingent derivative for set-valued map by using the contingent cone [1]. Corley established the sufficient and necessary optimality conditions for set-valued optimization problems by virtue of the concept of contingent and circatangent derivative [13]. The contingent derivative play an important role for set-valued optimization problems. But, necesary and sufficient optimality conditions do not coincide unified under standart assumptions. To overcome the difficulty, another of differentiability concept which is based on using epigraphs of set-valued maps was proposed Jahn and Rauh [23]. Kasımbeyli introduced in [24] the notion of the radial epiderivative of a nonconvex set-valued map. This definition of the radial epiderivative given by Kasımbeyli is different from that of Flores- Bazan [7] and is similar to the definition of the contingent epiderivative given by Jahn and Rauh [23]. He derived the formulation of optimality conditions in the single valued and set-valued optimization without convexity assumption and investigated relationships between this kind of epiderivative and weak subdifferentials and directional derivatives for real-valued nonconvex functions. Kasımbeyli and Inceoglu introduced in [25] the notion of generalized radial epiderivative for set-valued maps and investigated existence conditions for generalized radial epiderivative. They established the relationship between the radial epiderivative and the generalized radial epiderivative. By using the generalized radial epiderivative, Kasımbeyli and Inceoglu presented the necessary and sufficient optimality conditions for set-valued optimization. Recently, there has been an increasing interest in second-order and higher-order optimality research for set-valued map [3, 4, 5, 6, 9, 11, 12, 14, 15, 16, 21, 22, 27, 20, 26, 29, 30, 32]. Jahn et al. proposed the second-order epiderivatives in terms of the second-order contingent set [21], introduced by Aubin and Frankowska [2]. They obtained the second-order optimality conditions by using these dervatives in set-valued optimization.It can be seen that a second-order contingent set, introduced by Aubin and Frankowska [2], and a second-order asymptotic contingent cone, introduced by Penot [30], play a important role in establishing second-order optimality conditions. Li et al. proposed a generalized second-order composed contingent epiderivative for a set-valued map and investigated some of its properties. By virtue of the generalized second-order composed contingent epiderivative, they also establised a unified second-order sufficient and c  2018 BISKA Bilisim Technology ∗ Corresponding author e-mail: inceoglugonca@gmail.com