# Moreau's decomposition theorem

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

## Revision as of 17:34, 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.

## 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*.

## Proof of extended Farkas' lemma

Let be arbitrary. Then, by Moreau's theorem we have

and

Therefore,

In particular, for any we have . Hence, . Similarly, for any we have . Hence, . Therefore, .