# Moreau's decomposition theorem

### From Wikimization

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== Projection on closed convex sets == | == Projection on closed convex sets == | ||

=== Projection mapping === | === Projection mapping === | ||

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

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=== Characterization of the projection === | === Characterization of the projection === | ||

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

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

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== Moreau's theorem == | == Moreau's theorem == | ||

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Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. | Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces. | ||

## Revision as of 11:39, 17 July 2009

Sándor Zoltán Németh

## 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 Let and be arbitrary. By using the convexity of it follows that 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.

**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 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: By using the characterization of the projection, we have for all In particular, if then and if then Thus, Denote Then, It remains 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

### notes

For definition of *convex cone* see Convex cone, Wikipedia; in finite dimension see Convex cones, Wikimization.

For definition of *polar cone* in finite dimension, see more at Dual cone and polar cone.

### 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.