Moreau's decomposition theorem
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and thus <math>y=P_{\mathcal K^\circ}z</math>.</li> | and thus <math>y=P_{\mathcal K^\circ}z</math>.</li> | ||
- | <li>2<math>\Rightarrow</math>1:</li> | + | <li>2<math>\Rightarrow</math>1: By the characterization of the projection we have |
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+ | <center> | ||
+ | <math> | ||
+ | \langle z-x,p-x\rangle\leq0, | ||
+ | </math> | ||
+ | </center> | ||
+ | |||
+ | for all <math>p\in\mathcal K</math>. In particular, if <math>p=0</math>, then | ||
+ | <math>\langle z-x,x\rangle\geq0</math> and if <math>p=2x</math>, then <math>\langle z-x,x\rangle\leq0</math>. Thus, <math>\langle z-x,x\rangle=0</math>. Denote <math>y=z-x</math>. | ||
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+ | We will show that <math>z-P_{\mathcal K}z=P_{\mathcal K^\circ}z</math>.</li> Indeed, let <math>q\in K^\circ</math> be arbitrary. Then, | ||
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+ | <center> | ||
+ | <math> | ||
+ | \langle z-(z-P_{\mathcal K}z),q-(z-P_{\mathcal K}z)\rangle | ||
+ | </math> | ||
+ | </center> | ||
+ | |||
</ul> | </ul> |
Revision as of 16:00, 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 we have
- 21: By the characterization of the projection we have
for all . In particular, if , then and if , then . Thus, . Denote .
We will show that . Indeed, let be arbitrary. Then,