The Subdifferential of a Convex Function

In this chapter, the first section presents the definitions of Gateaux differentiable functions and of Frechet differentiable functions and the concept of subdifferentiability. Monotone and maximal monotone operators are defined. Minty’s theorem is proved. The subdifferential is shown to be a maximal monotone operator. The conjugate function is used to transform a minimization problem into a maximization problem and conversely. Finally, the additivity of the subdifferential is studied.