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| In [[convex analysis]], '''Danskin's theorem''' is a [[theorem]] which provides information about the [[derivative]]s of a [[function (mathematics)|function]] of the form
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| :<math>f(x) = \max_{z \in Z} \phi(x,z).</math>
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| The theorem has applications in [[optimization (mathematics)|optimization]], where it sometimes is used to solve [[minimax]] problems.
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| == Statement ==
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| The theorem applies to the following situation. Suppose <math>\phi(x,z)</math> is a [[continuous function]] of two arguments,
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| :<math>\phi: {\mathbb R}^n \times Z \rightarrow {\mathbb R}</math>
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| where <math>Z \subset {\mathbb R}^m</math> is a [[compact set]]. Further assume that <math>\phi(x,z)</math> is [[convex function|convex]] in <math>x</math> for every <math>z \in Z</math>. | |
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| Under these conditions, Danskin's theorem provides conclusions regarding the [[differentiability]] of the function
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| :<math>f(x) = \max_{z \in Z} \phi(x,z).</math>
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| To state these results, we define the set of maximizing points <math>Z_0(x)</math> as
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| :<math>Z_0(x) = \left\{ \overline{z} : \phi(x,\overline{z}) = \max_{z \in Z} \phi(x,z)\right\}.</math>
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| Danskin's theorem then provides the following results.
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| ;Convexity
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| : <math>f(x)</math> is [[convex function|convex]].
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| ;Directional derivatives
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| : The [[directional derivative]] of <math>f(x)</math> in the direction <math>y</math>, denoted <math>D_y\ f(x)</math>, is given by | |
| ::<math>D_y f(x) = \max_{z \in Z_0(x)} \phi'(x,z;y),</math>
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| : where <math>\phi'(x,z;y)</math> is the directional derivative of the function <math>\phi(\cdot,z)</math> at <math>x</math> in the direction <math>y</math>. | |
| ;Derivative
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| : <math>f(x)</math> is [[differentiable]] at <math>x</math> if <math>Z_0(x)</math> consists of a single element <math>\overline{z}</math>. In this case, the [[derivative]] of <math>f(x)</math> (or the [[gradient]] of <math>f(x)</math> if <math>x</math> is a vector) is given by
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| :: <math>\frac{\partial f}{\partial x} = \frac{\partial \phi(x,\overline{z})}{\partial x}.</math>
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| ;Subdifferential
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| :If <math>\phi(x,z)</math> is differentiable with respect to <math>x</math> for all <math>z \in Z</math>, and if <math>\partial \phi/\partial x</math> is continuous with respect to <math>z</math> for all <math>x</math>, then the [[subdifferential]] of <math>f(x)</math> is given by
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| :: <math>\partial f(x) = \mathrm{conv} \left\{ \frac{\partial \phi(x,z)}{\partial x} : z \in Z_0(x) \right\}</math>
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| : where <math>\mathrm{conv}</math> indicates the [[convex hull]] operation.
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| == See also ==
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| * [[Maximum theorem]]
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| * [[Envelope theorem]]
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| * [[Hotellings Lemma]]
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| == References ==
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| * {{cite book
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| | last = Bertsekas
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| | first = Dimitri P.
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| | title = Nonlinear Programming
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| | publisher = Athena Scientific
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| | date = 1999
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| | pages = 717
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| | location = Belmont, MA
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| | id = ISBN 1-886529-00-0 }}
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| [[Category:Convex analysis]]
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| [[Category:Mathematical optimization]]
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| [[Category:Theorems in analysis]]
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| [[Category:Convex optimization]]
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