To give some context: I am aware of the uses of Convex Analysis (and its applications in Convex Optimization), I have been studying (for a while) the developments of Nonsmooth Analysis (and its applications in Nonsmooth Optimization) as traced by Frank Clarke.
While all of this work on subdifferentials and subgradients is very interesting, I am wondering about its relevance in mathematical research. I have been told some nonsmooth optimization methods have been narrowly applied in Engineering (neural nets, robotics, dynamics, ...) but I am wondering how the field connects with the rest of mathematical research.
Thank you for the insight.