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>. </li> |
<li>1<math>\Rightarrow</math>2</li> | <li>1<math>\Rightarrow</math>2</li> | ||
</ul> | </ul> |
Revision as of 15:31, 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 .
- 12