Moreau's decomposition theorem
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== Characterization of the projection == | == Characterization of the projection == | ||
- | Let <math>\mathcal | + | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space, <math>\mathcal C</math> a closed convex set in <math>\mathcal H,\,u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, <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>. |
== Proof == | == Proof == |
Revision as of 01:53, 11 July 2009
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Characterization of the projection
Let be a Hilbert space, a closed convex set in and . Then, 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.