# Moreau's decomposition theorem

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

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## Revision as of 02:29, 11 July 2009

## Contents |

## Projection mapping

Let be a Hilbert space and a closed convex set in . The **projection mapping** onto is the mapping defined by and

## Characterization of the projection

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.