Techniques for Nonsmooth Analysis on Smooth Manifolds I: Local Problems

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Nonsmooth analysis, differential analysis for functions without differentiability, has witnessed a rapid growth in the past several decades stimulated by intrinsic nonsmooth phenomena in control theory, optimization, mathematical economics and many other fields. In the past several y ears many problems in control theory, matrix analysis and geometry naturally led to an increasing interest in nondifferentiable functions on smooth manifolds. Since a smooth manifold only locally resembles a Euclidean space and, in general, lacks of a linear structure, new techniques are needed to adequately address these problems. A number of results and techniques for dealing with such problems have emerged recently [8, 16, 18, 32]. The purpose of this paper is to report some useful techniques that we developed in the past several y ears for studying nonsmooth functions on smooth manifolds.