# Moreau's decomposition theorem

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* J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962. | * J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962. | ||

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- | == Extended Farkas' lemma == | ||

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- | For any closed convex cone <math>\mathcal J</math> in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math>, denote by <math>\mathcal J^\circ</math> the polar cone of <math>\mathcal J</math>. Let <math>\mathcal K</math> be an arbitrary closed convex cone in <math>\mathcal H</math>. Then, the extended Farkas' lemma asserts that <math>\mathcal K^{\circ\circ}=\mathcal K.</math> Hence, denoting <math>\mathcal L=\mathcal K^\circ,</math> it follows that <math>\mathcal L^\circ=\mathcal K</math>. Therefore, the cones <math>\mathcal K</math> and <math>\mathcal L</math> are called ''mutually polar pair of cones''. | ||

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- | == Proof of extended Farkas' lemma == | ||

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- | (Sándor Zoltán Németh) Let <math>z\in\mathcal H</math> be arbitrary. Then, by Moreau's theorem we have | ||

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- | <center> | ||

- | <math> | ||

- | z=P_{\mathcal K}z+P_{\mathcal K^\circ}z | ||

- | </math> | ||

- | </center> | ||

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- | and | ||

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- | <center> | ||

- | <math> | ||

- | z=P_{\mathcal K^\circ}z+P_{\mathcal K^{\circ\circ}}z. | ||

- | </math> | ||

- | </center> | ||

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- | Therefore, | ||

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- | <center> | ||

- | <math> | ||

- | P_{\mathcal K^{\circ\circ}}z=P_{\mathcal K}z=z-P_{\mathcal K^\circ}z. | ||

- | </math> | ||

- | </center> | ||

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- | In particular, for any <math>z\in K</math> we have <math>\mathcal K^{\circ\circ}\ni P_{\mathcal K^{\circ\circ}}z=z</math>. Hence, <math>\mathcal \mathcal K^{\circ\circ}\supset K</math>. Similarly, for any <math>z\in K^{\circ\circ}</math> we have <math>z= P_{\mathcal K}z\in\mathcal K</math>. Hence, <math>\mathcal K^{\circ\circ}\subset\mathcal K</math>. Therefore, <math>\mathcal K^{\circ\circ}=\mathcal K</math>. |

## Revision as of 20:28, 10 July 2009

**Moreau's theorem** is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.

Let be a closed convex cone in the Hilbert space and its polar. For an arbitrary closed convex set in , denote by the projection onto . For the following statements are equivalent:

- and
- and

## Proof of Moreau's theorem

Let be an arbitrary closed convex set in and . Then, it is well known that if and only if for all . We will call this result the * characterization of the projection*.

- 12: For all we have
.

Then, by the characterization of the projection, it follows that . Similarly, for all we have

- 21: Let . By the characterization of the projection we have for all . In particular, if then and if then . Thus, . Denote . Then, . It remained to show that . First, we prove that . For this we have to show that , for
all . By using the characterization of the projection, we have
for all . Thus, . We also have

for all , because . By using again the characterization of the projection, it follows that .

## References

- J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962.