Subdifferential Calculus

The subject matter of this chapter is a calculus governing the limiting subdifferentials of composite functions, that is functions that are composed from lower semi-continuous extended valued functions and indicator functions of closed sets. It is remarkable the extent to which a subdifferential calculus can be developed, reproducing aspects of classical calculus governing differential properties of smooth functions, when the functions involved are no longer smooth. A distinctive feature of subdifferential calculus is that, typically, it does not provide precise representations of the limiting subdifferentials of composite functions, but only estimates sets for these limiting subdifferentials. We have already encountered a nonsmooth mean value theorem in the previous chapter. The subdifferential calculus presented in this chapter provides versions of the sum rule, and chain rule. It also provides a widely used ‘max rule’ for composite functions, which has no parallel classical calculus. To give a foretaste of techniques employed in later chapters, we derive a very general Lagrange multiplier in nonlinear programming, using subdifferential calculus in harness with a variational principle.