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It is obvious that there is similarity between subgradient and gradient. The subgradient of smooth functions is reduced to gradient. I have two questions.

The first is does there exist subgradient decent method (analogous to the well known gradient decent mothod) in nonsmooth problems.

The second is, given a nonsmooth function, how to determine the subgradient? (by computer)

Summarily, I have no idea of the role of subgradient in programming of nonsmooth functions. Is it just a concept or a very useful tool (like gradient)?

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There is an obvious Google search here ("subgradient descent")... –  Qiaochu Yuan Apr 2 '13 at 6:38
this is not a research level MO question; please have a look at any of the textbooks on nonsmooth stuff; for convex optimization: "introductory lectures on convex optimization" by Yu. Nesterov; for nonconvex, nonsmooth functions, subdifferential calculus is much more tricky, but a practical starting point it "Nonsmooth analysis" by F. Clarke; also, have a look at online slides of S. Boyd's 364b course at stanford... –  Suvrit Apr 2 '13 at 6:50

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