Nonsmooth analysis and its applications on Riemannian manifolds

1. Nonsmooth analysis and its applications on Riemannian manifolds S. Hosseini FSDONA 2011, Germany.

2. Nonsmooth analysis

3. MotivationNonsmoothFunctions are often considered on Euclidean spaces! • Unlike Euclidean spaces, a manifold in general does not have a linear structure! • However, in many aspects of mathematics such as control theory and matrix analysis, problems arise on smooth manifolds!

4. Therefore, new techniques are needed for dealing with problems on manifolds! • 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. • 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:

5. Question; How can we deal with general problems? Such as the existence of solutions and necessary conditions of optimality for a general problem.

7. Our key tools; • Palais -Smale condition; • Ekelandvariational principle;

8. Palais-Smale condition,

10. Definitions

11. Subderivative of lower semi continuous functions on Riemannian manifolds

12. Contigent derivative of lower semi continuous functions on Riemannian manifolds

13. Generalized directional derivative;

14. Generalized gradient

15. Generalization of classical derivative

16. New Definitions

17. Bouligand tangent cone

18. Clarke tangent cone

20. Clarke normal cone

21. Palais-Smale condition

22. Our main results

23. Lebourgs Mean Value Theorem

29. Applications of nonsmooth analysis on manifolds

30. Optimization

34. Characterization of epi-Lipschitz subset of Rimannian manifold

35. Epi-Lipschitz subsets of Rimannian manifolds

37. Applications of Epi-Lipschitz subsets

39. Noncritical neck principal of Morse theory