Subderivatives and Subgradients

Maximization and minimization are often useful in constructing new functions and mappings from given ones, but, in contrast to addition and composition, they commonly fail to preserve smoothness. These operations, and others of prime interest in variational analysis, fit poorly in the traditional environment of differential calculus. The conceptual platform for ‘differentiation’ needs to be enlarged in order to cope with such circumstances.

Notions of semidifferentiability and epi-differentiability have already been developed in Chapter 7 as a start to this project. The task is carried forward now in a thorough application of the variational geometry of Chapter 6 to epigraphs. ‘Subderivatives’ and ‘subgradients’ are introduced as counterparts to tangent and normal vectors and shown to enjoy various useful relationships. Alongside of general subderivatives and subgradients, there are ‘regular’ ones of more special character. These are intimately tied to the regular tangent and normal vectors of Chapter 6 and show aspects of convexity. The geometric paradigm of Figure 6–17 finds its reflection in Figure 8–9, which schematizes the framework in which all these entities hang together.