# Moreau's decomposition theorem

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- | <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>. Then, by the characterization of the projection it follows that <math>x=P_{\mathcal K}z</math>. </li> | + | <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>. Then, by the characterization of the projection it follows that <math>x=P_{\mathcal K}z</math>. Similarly, for all <math>q\in K^\circ</math>\langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0</math> and thus <math>y=P_{\mathcal K^\circ}z</math>.</li> |

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

</ul> | </ul> |

## Revision as of 15:36, 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 \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0</math> and thus .
- 12