Abstract
In the paper, we give some remarks on [1]. Then, we modify main results concerning the sum rule of second-order contingent derivatives for set-valued maps and its application to the sensitivity analysis of generalized perturbation maps. The obtained results are new and better than those in [1]. Some examples are proposed to illustrate our results.
Highlights
In [1], the second-order proto-differentiability and second-order semi-differentiability for set-valued maps were firsthy discussed and applied to sum rules of two set-valued maps
The semidifferentiability plays an essential role in all main results in [1]
A new result is proposed to avoid the semidifferentiability by using a weaker hypothesis of the proto-differentiability
Summary
In [1], the second-order proto-differentiability and second-order semi-differentiability for set-valued maps were firsthy discussed and applied to sum rules of two set-valued maps. The second-order lower Dini derivative of F at (x, y) in direction (w, r) is a set-valued map Dl2F(x, y, w, r) : X 2Y that is defined by (ii) The map F is said to be second-order semidifferentiable at x relative to y in the direction (w, r) if D2F(x, y, w, r) Dl2F(x, y, w, r) .
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