On Derivatives Of The Generalized Perturbation Maps

This paper is mainly concerned with the derivatives of the generalized perturbation maps defined by two set-valued maps and a set. The expression of the derivatives of the generalized perturbation maps are given. Here, the approach to the differentiation of a set-valued map comes from the classical geometric view, which regards the graph of the derivative at a point as the tangent cone or norm cone to the graph of the set-valued map. It contains the tangent derivatives (coderivatives) which defined by the tangent cones (normal cones) of the graph of the set-valued maps at the consideration point. The detailed contents are listed below:In finite dimensional spaces, we discuss derivatives,which are defined by tangent cones of graphs of the generalized perturbation maps. Firstly, we transform the problem for solving derivatives of the generalized perturbation maps into the problem for solving derivatives of sum of two set-valued maps. Then, by using the exact calculus rules of derivatives of sum of two set-valued maps in [28], we obtain derivatives of the generalized perturbation maps. Under weaker conditions, we obtain the expressions of the contingent and adjacent derivatives of the generalized perturbation maps, respectively. Furthermore, we show the generalized perturbation maps to be proto-differentiable when the two set-valued maps are proto-differentiable and obtain their calculus formulations. In the end, we give some corollaries and the expression of the derivatives under the special cases. Simultaneously, we use some examples to illustrate reasonability of the assumptions of these theorems and effectiveness of these conclusions.In finite dimensional spaces, we studied the calculus rules of the coderivatives, which is defined by norm cone of graph of the generalized perturbation maps. Under a weaker constraint qualification than the normal constraint qualification in [38], we proved an inclusion relation between coderivative of the composition for two set-valued maps and the composition of coderivative of two set-valued maps, which is a generalization of Theorem 3.13 in [38]. Then, we studied the exact calculus rules of coderivative of the composition for two set-valued maps under some Lipschitzian conditions. Simultaneously, some examples are given to illustrate reasonability of the assumptions of these theorems and effectiveness of these conclusions. Soon after, as a special case of composition, we obtained the inclusion relations and the exact expressions of the coderivatives of sum of two set-valued maps. In the end, by using the calculus rules of coderivatives of two set-valued maps, we obtain exact calculus rules of the coderivatives of generalized perturbation maps and two corollaries.