nonsmooth analysis and its applications on riemannian manifolds
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Nonsmooth analysis and its applications on Riemannian manifolds. S. Hosseini FSDONA 2011, Germany. Nonsmooth analysis. Motivation Nonsmooth Functions are often considered on Euclidean spaces!. Unlike Euclidean spaces, a manifold in general does not have a linear structure!. - PowerPoint PPT PresentationTRANSCRIPT
Nonsmooth analysis and its applications on Riemannian
manifoldsS. Hosseini
FSDONA 2011, Germany.
Nonsmooth analysis
However, in many aspects of mathematics such as control theory and matrix analysis, problems arise on smooth manifolds!
Unlike Euclidean spaces, a manifold in general does not have a linear structure!
MotivationNonsmooth Functions are often considered on Euclidean spaces!
A useful technique in dealing with the problems on Riemannian manifolds that are local is by using known result in an Euclidean space along the following line:
1. Convert the problem into one in an Euclidean space.
2. Apply corresponding result in an Euclidean space to the problem.
3. Lift the conclusion back onto the manifold.
Therefore, new techniques are needed for dealing with problems on manifolds!
Question; How can we deal with
general problems?Such as the existence of solutions and necessary conditions of optimality for a general problem.
Ekeland variational principle;
Palais -Smale condition;
Our key tools;
Palais-Smale condition,
For smooth functions on Rimannian manifolds
K. C. Chang, For locally Lipschitz functions on Hilbert spaces,1981.
E. Mirenchi, A. Salvatore, For locally Lipschitz
functions on Riemannian manifolds,1994.
K. Le, D. Motreanu, For lower semi continuos functions on Hilbert
spaces,2002.
S. Hosseini, M.R. Pouryayevali, For lower
semi continuous functions on Riemannian
manifolds,2010.
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Definitions
Subderivative of lower semi continuous functions on Riemannian manifolds
Contigent derivative of lower semi continuous functions on Riemannian manifolds
Generalized directional derivative;
Generalized gradient
Generalization of classical derivative
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New Definitions
Bouligand tangent cone
Clarke tangent cone
Clarke normal cone
Palais-Smale condition
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Our main results
Lebourgs Mean Value Theorem
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Applications of nonsmooth analysis on manifolds
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Optimization
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Characterization of epi-Lipschitz subset of Rimannian manifold
Epi-Lipschitz subsets of Rimannian manifolds
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Applications of Epi-Lipschitz subsets
Noncritical neck principal of Morse theory