Generalized Directional Derivatives and Subgradients of Nonconvex Functions

TLDR

A generalized differentiation theory associates with any extended‑real function on a linear topological space a convex, weak*‑closed set of subgradients, forming the sub‑differential mapping ∂f. The sub‑differential calculus provides rules linking ∂f to generalized directional derivatives and to sub‑differentials of component functions, enabling its computation and estimation.

Abstract

Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ : x →∂ƒ( x ) is the sub differential of ƒ. Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ 1 + ƒ 2 ,ƒ = g o A, etc.), comprise the sub differential calculus.