I saw a tutorial/expository journal article a while ago that focused on introducing intuitively different notions of subdifferentials appropriate for general nonlinear optimization. I forgot the journal and title. Any suggestions?

## 2 Answers

Section 1.3 of Chapter *Nonsmooth Optimization* by V. F. Demyanov in the book Nonlinear Optimization discusses in some detail the Dini, Hadamard, Shor, Clarke, and Michel–Penot notions of the subdifferential (in addition to the most common notion of the subdifferential, for convex functions, given in Section 1.2.2 of the chapter). Remark 1.3.1 on p. 72 of that chapter contains a number of references to "[m]any other subdifferentials".

I found the article: J. Li, A. M. -C. So and W. -K. Ma, "Understanding Notions of Stationarity in Nonsmooth Optimization: A Guided Tour of Various Constructions of Subdifferential for Nonsmooth Functions," in IEEE Signal Processing Magazine, vol. 37, no. 5, pp. 18-31, Sept. 2020, doi: 10.1109/MSP.2020.3003845., and the corresponding arxiv preprint.

It reviews subdifferentials for convex functions, then uses them as the basis of providing a list of desiderata for generalized subdifferentials. Then it moves on to subdifferentials for locally Lipschitz functions-- the Bouligand and then Clarke subdifferentials. Then it moves on to Frechet subdifferentials, the limiting subdifferential, and the relationships between these and the Clarke subdifferential. Finally it uses these notions to discuss stationarity.

The upside of this article for me is that it liberally peppers examples of applications of these notions to signal processing applications throughout, so the coverage and development of these subdifferentials is not as abstract as elsewhere in the more mathematical literature.