# Moreau's decomposition 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. | ||

- | 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. | + | 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. For an arbitrary closed convex set <math>\mathcal C</math> in <math>\mathcal H</math>, denote by <math>P_{\mathcal C}</math> the projection onto <math>\mathcal C</math>. For <math>x,y,z\in\mathcal H</math> the following two statements are equivalent: |

<ol> | <ol> | ||

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</li> | </li> | ||

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

+ | |||

+ | == Proof == | ||

+ | |||

+ | Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H</math>, <math>u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, it is well known that <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>. We will call this result the '''''characterization of the projection'''''. | ||

+ | |||

+ | <ul> | ||

+ | <li>1<math>\Rightarrow</math>2: For all <math>p\in K</math> we have <math>\langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0</math></li> | ||

+ | <li>1<math>\Rightarrow</math>2</li> | ||

+ | </ul> |

## Revision as of 15:25, 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
- 12