Moreau's decomposition theorem
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== Proof == | == Proof == | ||
- | Suppose that <math>v=P_{\mathcal C}u</math> and let <math>w\in\mathcal C</math> be arbitrary. By | + | Suppose that <math>v=P_{\mathcal C}u</math> and let <math>w\in\mathcal C</math> be arbitrary. By using the convexity of <math>\mathcal C</math>, it follows that <math>(1-t)v+tw\in\mathcal C</math>, for all <math>t\in (0,1)</math>. Then, by using the definition of the projection, we have |
- | the convexity of <math>\mathcal C</math> it follows that <math>(1-t)v+tw\in\mathcal C</math>, for | + | |
- | all <math>t\in (0,1)</math>. Then, by the definition of the projection we have | + | |
<center> | <center> | ||
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</center> | </center> | ||
- | By tending with <math>t</math> to <math>0</math> we get <math>\langle u-v,w-v\rangle\leq0</math>. | + | By tending with <math>t</math> to <math>0</math>, we get <math>\langle u-v,w-v\rangle\leq0</math>. |
+ | <br> | ||
+ | <br> | ||
+ | |||
+ | Conversely, suppose that <math>\langle u-v,w-v\rangle\leq0,</math> for all <math>w\in\mathcal C</math>. Then, | ||
+ | |||
+ | <center> | ||
+ | <math>\|u-w\|^2=\|u-v-(w-v)\|^2=\|u-v\|^2-2\langle u-v,w-v\rangle+\|w-v\|^2\geq \|u-v\|^2,</math> | ||
+ | </center> | ||
+ | |||
+ | for all <math>w\in\mathcal C</math>. Hence, by using the definition of the projection, we get <math>v=P_{\mathcal C}u</math>. | ||
== Moreau's theorem == | == Moreau's theorem == |
Revision as of 01:48, 11 July 2009
Contents |
Characterization of the projection
Let be an arbitrary closed convex set in and . Then, it is well known that if and only if for all .
Proof
Suppose that and let be arbitrary. By using the convexity of , it follows that , for all . 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.
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 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: Let . By the characterization of the projection we have for all . In particular, if then and if then . Thus, . Denote . Then, . It remained 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 .
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.