# Moreau's decomposition theorem

### From Wikimization

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- | = Projection on closed convex sets = | + | == Projection on closed convex sets == |

- | == Projection mapping == | + | === Projection mapping === |

Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space and <math>\mathcal C</math> a closed convex set in <math>\mathcal H</math>. The '''projection mapping''' <math>P_{\mathcal C}</math> onto <math>\mathcal C</math> is the mapping <math>P_{\mathcal C}:\mathcal H\to\mathcal H</math> defined by <math>P_{\mathcal C}(x)\in\mathcal C</math> and | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space and <math>\mathcal C</math> a closed convex set in <math>\mathcal H</math>. The '''projection mapping''' <math>P_{\mathcal C}</math> onto <math>\mathcal C</math> is the mapping <math>P_{\mathcal C}:\mathcal H\to\mathcal H</math> defined by <math>P_{\mathcal C}(x)\in\mathcal C</math> and | ||

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</center> | </center> | ||

- | == Characterization of the projection == | + | === Characterization of the projection === |

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Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space, <math>\mathcal C</math> a closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, <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>. | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space, <math>\mathcal C</math> a closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, <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>. | ||

- | == Proof == | + | === Proof === |

Suppose that <math>v=P_{\mathcal C}u</math> and let <math>w\in\mathcal C</math> be arbitrary. By using the convexity of <math>\mathcal C</math>, it follows that <math>(1-t)v+tw\in\mathcal C</math>, for all <math>t\in (0,1)</math>. Then, by using the definition of the projection, we have | Suppose that <math>v=P_{\mathcal C}u</math> and let <math>w\in\mathcal C</math> be arbitrary. By using the convexity of <math>\mathcal C</math>, it follows that <math>(1-t)v+tw\in\mathcal C</math>, for all <math>t\in (0,1)</math>. Then, by using the definition of the projection, we have | ||

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for all <math>w\in\mathcal C</math>. Hence, by using the definition of the projection, we get <math>v=P_{\mathcal C}u</math>. | for all <math>w\in\mathcal C</math>. Hence, by using the definition of the projection, we get <math>v=P_{\mathcal C}u</math>. | ||

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

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</ol> | </ol> | ||

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

<ul> | <ul> | ||

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</ul> | </ul> | ||

- | == References == | + | === 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. | * J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962. |

## Revision as of 06:05, 11 July 2009

## Contents |

## Projection on closed convex sets

### 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.