Smooth criminals (or if you prefer, smooth approximations of non-smooth functions)

Abstract: Although smooth functions are much easier to work with than non-smooth ones, from an optimization viewpoint we are often highly interested in points where a function is NOT smooth. Think, for example, of the absolute value function, whose minimum occurs at the origin. This leads to a major distinction between optimization and (general) analysis: even if a function is smooth a.e., we cannot simply ignore its non-smooth points. One way to unify the discussion is to talk about smooth approximations of non-smooth functions, such as the Moreau envelope. I will introduce such approximations, talk about some of their analytic properties, and discuss how they are used in algorithm development and analysis.*

 

*I won't assume any prior knowledge or optimization or algorithms, so the talk should be accessible to everyone!