# Moreau's decomposition theorem

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

- | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space, <math>\mathcal C</math> a closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in | + | For any closed convex set <math>\mathcal D</math> in <math>\mathcal H</math>, denote by <math>P_{\mathcal D}</math> the '''projection mapping onto <math>\mathcal D</math>'''; that is, the mapping <math>P_{\mathcal D}:\mathcal H\to\mathcal H</math> defined by <math>P_{\mathcal D}(x)\in\mathcal D</math> and |

+ | |||

+ | <center> | ||

+ | <math>\|x-P_{\mathcal D}(x)\|=\min\{\|x-y\|:y\in\mathcal D\}.</math> | ||

+ | </center> | ||

+ | |||

+ | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space, <math>\mathcal C</math> a closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, <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 == | == Proof == |

## Revision as of 01:18, 11 July 2009

## Contents |

## Characterization of the projection

For any closed convex set in , denote by the **projection mapping onto **; that is, the mapping defined by and

Let be a Hilbert space, a closed convex set in and . Then, if and only if for all .

## Proof

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

.

Hence,

By tending with to , we get .

Conversely, suppose that for all . Then,

for all . Hence, by using the definition of the projection, 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 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.