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.

  • $\begingroup$ See Heinonen, Nonsmooth Calculus. Connections to other areas and applications are discussed in the last section 16. $\endgroup$
    – Conifold
    Commented Dec 21, 2019 at 22:42
  • 1
    $\begingroup$ @Conifold: I believe this is unrelated to what the OP is looking for. Distributional derivatives are quite different to the generalised gradients and subgradients of Clarke's analysis. $\endgroup$
    – copper.hat
    Commented Dec 22, 2019 at 3:12
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    $\begingroup$ An application area I found to be interesting was that of total variation denoising, which has some properties that are preferred over L2-norm denoising but yields a convex optimization problem, which can be solved using subgradient methods. $\endgroup$
    – Mikal
    Commented Mar 11, 2020 at 9:53
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    $\begingroup$ Compressed sensing, which solves the LASSO problem, is huge! It has revolutionized MRI. $\endgroup$
    – NicNic8
    Commented Mar 29, 2020 at 16:36


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