@MISC{Yu12thedifferentiability, author = {Yao-liang Yu}, title = {The Differentiability of the Upper Envelop}, year = {2012} }

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Abstract

We present the proof of the Danskin-Valadier theorem, i.e. when the directional derivative of the supremum of a collection of functions admits a natural representation. 1 Preliminary Consider a collection of extended real-valued functions fi: X 7 → R̄, where i ∈ I is some index set, X is some real vector space, and R ̄: = R ∪ {±∞}. Define the supremum (i.e. upper envelop) of the collection as f(x): = sup i∈I fi(x). (1) We are interested in studying the directional derivative of f, hopefully relating it to the directional derivatives of fi. Recall that the directional derivative of g, along direction d, is defined as g′(x; d): = lim t↓0 g(x+ td) − g(x)