Abstract
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.
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© 2006 Springer Science+Business Media, Inc.
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(2006). Nonsmooth Optimization. In: Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-31256-9_6
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DOI: https://doi.org/10.1007/978-0-387-31256-9_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-29570-1
Online ISBN: 978-0-387-31256-9
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