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 years 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 years for studying nonsmooth functions on smooth manifolds.
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