Set-valued differentiation as an operator

Serguei Samborski

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Set-valued differentiation as an operator

Serguei Samborski

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Abstract

We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient may be extended in these spaces as a closed operator. If a function f belongs to the domain of such extension, then f is locally lipschitzian and the values of extended gradient coincide with the values of Clarke's gradient. However, unlike Clarke's gradient, our generalized gradient is a linear operator.

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Serguei Samborski. 2005-09-07. Set-valued differentiation as an operator. https://arxiv.org/abs/math/0509167

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