Moreau's decomposition theorem
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=== Projection mapping === | === Projection mapping === | ||
- | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space and <math>\mathcal C</math> a closed convex set in <math>\mathcal H</math> | + | Let <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> be a Hilbert space and <math>\mathcal C</math> a closed convex set in <math>\mathcal H.</math> The '''projection mapping''' <math>P_{\mathcal C}</math> onto <math>\mathcal C</math> is the mapping <math>P_{\mathcal C}:\mathcal H\to\mathcal H</math> defined by <math>P_{\mathcal C}(x)\in\mathcal C</math> and |
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- | 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> | + | 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 === | ||
- | Suppose that <math>v=P_{\mathcal C}u</math> | + | Suppose that <math>v=P_{\mathcal C}u.</math> Let <math>w\in\mathcal C</math> and <math>t\in (0,1)</math> be arbitrary. By using the convexity of <math>\mathcal C,</math> it follows that <math>(1-t)v+tw\in\mathcal C.</math> Then, by using the definition of the projection, we have |
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<math> | <math> | ||
- | \|u-v\|^2\leq\|u-[(1-t)v+tw]\|^2=\|u-v-t(w-v)\|^2=\|u-v\|^2-2t\langle u-v,w-v\rangle+t^2\|w-v\|^2 | + | \|u-v\|^2\leq\|u-[(1-t)v+tw]\|^2=\|u-v-t(w-v)\|^2=\|u-v\|^2-2t\langle u-v,w-v\rangle+t^2\|w-v\|^2, |
- | </math> | + | </math> |
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- | By tending with <math>t</math> to <math>0</math> | + | By tending with <math>t\,</math> to <math>0,\,</math> we get <math>\langle u-v,w-v\rangle\leq0.</math> |
<br> | <br> | ||
<br> | <br> | ||
- | Conversely, suppose that <math>\langle u-v,w-v\rangle\leq0,</math> for all <math>w\in\mathcal C</math> | + | Conversely, suppose that <math>\langle u-v,w-v\rangle\leq0,</math> for all <math>w\in\mathcal C.</math> Then, |
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</center> | </center> | ||
- | for all <math>w\in\mathcal C</math> | + | 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 05:02, 12 July 2009
Contents |
Projection on closed convex sets
Projection mapping
Let be a Hilbert space and a closed convex set in The projection mapping onto is the mapping defined by and
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 Let and be arbitrary. By using the convexity of it follows that 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. Recall that a convex cone in a vector space is a set which is invariant under the addition of vectors and multiplication of vectors by positive scalars (see more at Convex cone, Wikipedia or for finite dimension at Convex cones, Wikimization).
Theorem (Moreau) Let be a closed convex cone in the Hilbert space and its polar cone; that is, the closed convex cone defined by (for finite dimension see more at Dual cone and polar cone; see also Extended Farkas' lemma). 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 using 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.