# Moreau's decomposition theorem

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

Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H</math>, <math>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'''''. | Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H</math>, <math>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'''''. |

## Revision as of 17:09, 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 two statements are equivalent:

- , and
- and

## Contents |

## 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.

## The extended Farkas' lemma

For any closed convex cone in the Hilbert space , denote by the polar cone of . Let be an arbitrary closed convex cone in . Then, the extended Farkas' lemma asserts that Hence, denoting it follows that . Therefore, the cones and are called *mutually polar pair of cones*.