Nonsmooth Optimization

From the perspective of optimization, the subdifferential ∂f(·) of a convex function f has many of the useful properties of the derivative. Some examples: it gives the necessary optimality condition 0 ∈ ∂f(x) when the point x is a (local) minimizer (Proposition 3.1.5); it reduces to {∇f(x)} when f is differentiable at x (Corollary 3.1.10); and it often satisfies certain calculus rules such as ∂(f + g)(x) = ∂ f(x) + ∂ g(x) (Theorem 3.3.5). For a variety of reasons, if the function f is not convex, the subdifferential ∂f(·) is not a particularly helpful idea. This makes it very tempting to look for definitions of the subdifferential for a nonconvex function. In this section we outline some examples; the most appropriate choice often depends on context.