# Moreau's decomposition theorem

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Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. Recall that a '''convex cone''' in a vector space is a set which is invariant | Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. Recall that a '''convex cone''' in a vector space is a set which is invariant | ||

- | under the addition of vectors and multiplication of vectors by positive scalars (see more at [http://en.wikipedia.org/wiki/Convex_cone Convex cone (Wikipedia)] or for finite dimension at [ | + | under the addition of vectors and multiplication of vectors by positive scalars (see more at [http://en.wikipedia.org/wiki/Convex_cone Convex cone (Wikipedia)] or for finite dimension at [[Convex cones | Convex cone (Wikimization)]]). |

- | '''Theorem (Moreau)''' Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>\mathcal K^\circ</math> its '''polar cone'''; that is, the closed convex cone defined by <math>K^\circ=\{a\in\mathcal H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}</math> (for finite dimension see more at [http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone Dual cone and polar cone]; see also [ | + | '''Theorem (Moreau)''' Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>\mathcal K^\circ</math> its '''polar cone'''; that is, the closed convex cone defined by <math>K^\circ=\{a\in\mathcal H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}</math> (for finite dimension see more at [http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone Dual cone and polar cone]; see also [[Farkas%27_lemma#Extended_Farkas.27_lemma | Extended Farkas' lemma]]). For <math>x,y,z\in\mathcal H</math> the following statements are equivalent: |

<ol> | <ol> |

## Revision as of 04:51, 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. Recall that a **convex cone** in a vector space is a set which is invariant
under the addition of vectors and multiplication of vectors by positive scalars (see more at Convex cone (Wikipedia) or for finite dimension at Convex cone (Wikimization)).

**Theorem (Moreau)** Let be a closed convex cone in the Hilbert space and its **polar cone**; that is, the closed convex cone defined by (for finite dimension see more at Dual cone and polar cone; see also Extended Farkas' lemma). 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.