Positivity (2016) 20:295–298
A note on “Higher-order optimality conditions
in set-valued optimization using Studniarski derivatives
and applications to duality” [Positivity. 18,
· Li Min
· Peng Zhenhua
Received: 6 February 2015 / Accepted: 12 July 2015 / Published online: 21 July 2015
© Springer Basel 2015
Abstract The note points out that the sufﬁciency of proposition 2.1 in Anh (Positivity
18:449–473, 2014) is erroneous and we provide an example to illustrate it. Also the
proof of proposition 2.2 in Anh (Positivity 18:449–473, 2014) is incorrect and we give
a new proof.
Keywords Generalized subconvexlike · Convex cone · Set-valued map
Mathematics Subject Classiﬁcation 32F17 · 46G05 · 90C29 · 90C46
The concept of generalized convexity plays an important role in operations research
and applied mathematics. Yang et al. [1,2] introduced the concepts of generalized cone
subconvexlike set-valued map and nearly cone-subconvexlike set-valued map. Sach 
introduced a new convexity notion for set-valued maps, called ic-cone-convexlikeness.
Xu and Song  obtained the following results: (a) when the ordering cone has non-
empty interior, ic-cone-convexness is equivalent to near cone-subconvexlikeness; (b)
when the ordering cone has empty interior, ic-cone-convexness implies near cone-
subconvexlikeness, a counter example is given to show that the converse implication
is not true.
This research was supported by the National Natural Science Foundation of China Grant (11461044), the
Natural Science Foundation of Jiangxi Province (20151BAB201027) and the Science and Technology
Foundation of the Education Department of Jiangxi Province(GJJ12010).
Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China