# Moreau's decomposition theorem

### From Wikimization

m (→The extended Farkas' lemma) |
(→The extended Farkas' lemma) |
||

Line 54: | Line 54: | ||

== The extended Farkas' lemma == | == The extended Farkas' lemma == | ||

- | For any closed convex cone <math>\mathcal J</math> in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math>, denote by <math>\mathcal J^\circ</math> the polar cone of <math>\mathcal J</math>. Let <math>\mathcal K</math> be an arbitrary closed convex cone in <math>\mathcal H</math>. Then, <math>\mathcal K^{\circ\circ}=\mathcal K.</math> Hence, denoting | + | For any closed convex cone <math>\mathcal J</math> in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math>, denote by <math>\mathcal J^\circ</math> the polar cone of <math>\mathcal J</math>. Let <math>\mathcal K</math> be an arbitrary closed convex cone in <math>\mathcal H</math>. Then, the extended Farkas' lemma asserts that <math>\mathcal K^{\circ\circ}=\mathcal K.</math> Hence, denoting <math>\mathcal L=\mathcal K^\circ,</math> it follows that <math>\mathcal L^\circ=\mathcal K</math>. Therefore, the cones <math>\mathcal K</math> and <math>\mathcal L</math> are called ''mutually polar pair of cones''. |

- | <math>\mathcal L=\mathcal K^\circ,</math> it follows that <math>\mathcal L^\circ=\mathcal K</math>. | + | |

- | Therefore, the cones <math>\mathcal K</math> and <math>\mathcal L</math> are called ''mutually polar pair of cones''. | + |

## Revision as of 17:07, 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

## Proof

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*.