Subdifferential Calculus

Numerous facts about functions f:ℝⁿ → ℝ¯ and mappings F:ℝⁿ → ℝm and S:ℝⁿ ⇉ ℝ^m have been developed in Chapters 7, 8, and 9 by way of the variational geometry in Chapter 6 and characterized through subdifferentiation. In order to take advantage of this body of results, bringing the theory down from an abstract level to workhorse use in practice, one needs to have effective machinery for determining subderivatives, subgradients, and graphical derivatives and coderivatives in individual situations. Just as in classical analysis, contemplation of ε's and δ's only goes so far. In the end, the vitality of the subject rests on tools like the chain rule.

In variational analysis, though, calculus serves additional purposes. While classically the calculation of derivatives can't proceed without first assuming that the functions to be differentiated are differentiable, the subdifferentiation concepts of variational analysis require no such preconditions. Their rules of calculation operate in inequality or inclusion form with little more needed than closedness or semicontinuity, and they give a means of establishing whether a differentiability property or Lipschitzian property is present or not.