250146 VO Variational Analysis and structure in descent systems and optimization (2017W)

After a crash introductory course in Nonsmooth Analysis, we shall focus on Nonsmooth Optimization problems enjoying a nice intrinsic structure. In these lectures we shall discuss two main paradigms: The Tame (semi-algebraic) paradigm -which is what is nowadays called Tame Optimization and the Convex paradigm. Interactions with continuous/discrete dynamical systems of descent time will be clarified.
A secondary aim of this course is to provide essential background and material for further research. During the lectures, some open problems will be eventually mentioned.

Contents (Course syllabus)

1. A quick survey of Nonsmooth Analysis
1.1 From smooth manifolds to tangent and normal cones
1.2 Subdifferentials and co-derivatives
1.3 Lipschitz functions, Clarke subdifferential
1.4A nonsmooth Morse-Sard theorem and applications

2.Tame variational analysis
2.1 Semialgebraic functions, o-minimal structures
2.2 Stratification vs Clarke subdiferential
2.3 Sard theorem for tame multivalued maps
2.4 Lojasiewicz inequality and generalizations

3. Asymptotic analysis of descent systems
3.1 Proximal algorithm steepest descent
3.2 Asymptotic analysis: convergence, length, Palis & De Melo example
3.3 Kurdyka’s desigularization: characterization and applications
3.4 From R. Thom’s conjecture to the non-oscillating conjecture.
3.5 Tame Sweeping process desingularizing co-derivatives.

4. The convex paradigm
4.1 A convex counterexample to Kurdyka’s desigularization
4.2 Asymptotic equivalence between continuous and discrete systems
4.3 Self-contracted curves, Mancelli-Pucci mean width technique
4.4 Snake-like curves: convergence and rectifiability.