Abstract: Desirable geometric structures are prevalent in nonsmooth optimization problems. In particular, partial smoothness associated with certain active manifolds is well studied: the property is known to hold generically for semialgebraic functions, and yields nice identification properties. We extend the generic partial smoothness result to semialgebraic set-valued mappings with small graphs, hence establishing finite identification properties for a broader class of problems. Beyond partial smoothness, another interesting type of structures are conservative gradient fields. It is a generalization of subdifferentials suitable for analyzing deep learning algorithms. Our work gives a characterization of conservative gradient fields with a simple form.

Tencent meeting ID: 870437565