The Tangent Derivatives Of Piecewise Linear Mappings

The tangent derivatives of nonsmooth mappings as a generalization of the derivative operators of smooth mappings is a basic concept in variational analysis and plays an important role in nonsmooth optimization.However,the complexity of non-smooth and tangent derivative construction makes it quite difficult to calculate and estimate the tangent derivative of a specific non-smooth map,which limits the application of tangent derivatives in practical problems.In this paper,we give the exact expression of the Bouligand tangent derivative of a piecewise linear mapping,and give the quantitative estimate of its Clarke tangent derivative.Based on Zheng and Yang's the representation theorem of for piecewise linear mappings,We show that the tangent derivative of piecewise linear mapping on an infinite dimensional normed space can be expressed by the composition of the tangent derivative of piecewise linear mapping on a finite dimensional space and a linear operator.