# Moreau's decomposition theorem

### From Wikimization

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+ | == Characterization of the projection == | ||

+ | |||

+ | Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, it is well known that <math>v=P_{\mathcal C}u</math> if and only if <math>\langle u-v,w-v\rangle\leq0</math> for all <math>w\in\mathcal C</math>. | ||

+ | |||

+ | == Proof == | ||

+ | |||

+ | Suppose that <math>v=P_{\mathcal C}u</math> and let <math>w\in\mathcal C</math> be arbitrary. By | ||

+ | the convexity of <math>\mathcal C</math> it follows that <math>(1-t)v+tw\in\mathcal C</math>, for | ||

+ | all <math>t\in (0,1)</math>. Then, by the definition of the projection we have | ||

+ | |||

+ | <center> | ||

+ | <math> | ||

+ | \|u-v\|^2\leq\|u-[(1-t)v+tw]\|^2=\|u-v-t(w-v)\|^2=\|u-v\|^2-2t\langle u-v,w-v\rangle+t^2\|w-v\|^2 | ||

+ | </math>. | ||

+ | </center> | ||

+ | |||

+ | Hence, | ||

+ | |||

+ | <center> | ||

+ | <math>\langle u-v,w-v\rangle\leq\frac t2\|w-v\|^2.</math> | ||

+ | </center> | ||

+ | |||

+ | By tending with <math>t</math> to <math>0</math> we get <math>\langle u-v,w-v\rangle\leq0</math>. | ||

+ | |||

+ | == Moreau's theorem == | ||

+ | |||

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

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== Proof of Moreau's theorem == | == Proof of Moreau's theorem == | ||

- | |||

- | Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, it is well known that <math>v=P_{\mathcal C}u</math> if and only if <math>\langle u-v,w-v\rangle\leq0</math> for all <math>w\in\mathcal C</math>. We will call this result the '''''characterization of the projection'''''. | ||

<ul> | <ul> |

## Revision as of 01:37, 11 July 2009

## Contents |

## Characterization of the projection

Let be an arbitrary closed convex set in and . Then, it is well known that if and only if for all .

## Proof

Suppose that and let be arbitrary. By the convexity of it follows that , for all . Then, by the definition of the projection we have

.

Hence,

By tending with to we get .

## Moreau's theorem

**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

- 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.