Piecewise smoothness, local invertibility, and parametric analysis of normal, maps

• 成果类型: Article
• 署名作者: Pang, JS; Ralph, D
• 署名单位: University of Melbourne
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.21.2.401
• 发表日期: 1996
• 页码: 401-426
• 关键词: variational-inequalities; sensitivity analysis; nonlinear programs; directional differentiability; complementarity-problems; generalized equations; nonsmooth equations; multipliers; STABILITY; THEOREM

摘要

This paper is concerned with properties of the Euclidean projection map onto a convex set defined by finitely many smooth, convex inequalities and affine equalities. Under a constant rank constraint qualification, we show that the projection map is piecewise smooth (PC1) hence B(ouligand)-differentiable, or directionally differentiable; and a relatively simple formula is given for the B-derivative. These properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC1 function in terms of its B-derivative. An extension of the implicit function theorem which does not require local uniqueness is also presented. Degree theory plays a major role in the analysis of both the locally unique case and its extension.