On generalized gradients and optimization
نویسنده
چکیده
There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in the sense of convex analysis [1], of generalized directional derivative functions that are canonically associated to the locally Lipschitz functions under consideration. The key point to note is that such generalized directional derivative functions are automatically convex, even when the original locally Lipschitz functions are not convex. Two important special cases of functions f that are locally Lipschitz near some point x0 ∈ R and their generalized gradients, denoted by ∂̄f(x0), are as follows: (i) f is continuously differentiable at x0 and (ii) f is convex on R and x0 ∈ int dom f . Then
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