Constructive Nonsmooth AnalysisMany practical problems can be described adequately only with the aid of nonsmooth functions. The book is mainly devoted to describing problems, tools and results in the area of Nonsmooth Analysis and Nondifferentiable Optimization, in particular those that can be easily transformed into practically applicable and computationally implementable ones. Most of the attention is paid to different generalizations of the concept of derivative (subdifferentials in Convex Analysis, the Clarke Subdifferential, Quasidifferentials, Codifferentials), Calculus of Quasidifferentials, Implicit Functions, Nonsmooth Extremal Problems. |
Contents
Preface | 5 |
Homogeneous approximations of functions sets and mappings | 14 |
56 | 37 |
Copyright | |
16 other sections not shown
Common terms and phrases
a₁ approximation Assume B₁ B₂ Clarke subdifferential coincides concave function cone continuously codifferentiable Convex Analysis convex compact sets convex function convex set definition denote descent direction df(x ðƒ(x Dini derivative direction g directional derivative directionally differentiable element equivalent Example exists f(x+A f(xo function f G₂(U Hadamard differentiable Hence holds implicit function implies inclusion inequality Lemma Let a function let us consider lim inf lim sup linear locally Lipschitz minimal minimum point Minkowski duality multi-valued mapping N₁ neighbourhood nonempty open set operation Optimization pair positively homogeneous problem Proof properties Proposition quasidifferentiable quasidifferentiable functions real number Remark sequence set X CR space steepest descent sublinear function Subsection sufficiently small superdifferential superlinear support function Theorem 3.1 twice continuously U₁ upper semicontinuous v₁ vector w₁ w₂ yields zero ακ