Moreau's decomposition theorem
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== The extended Farkas' lemma == | == The extended Farkas' lemma == | ||
- | For any closed convex cone <math>\ | + | For any closed convex cone <math>\mathcal M</math> in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math>, denote by <math>\mathcal M^\circ</math> the polar cone of <math>\mathcal M</math>. Let <math>\mathcal K</math> be an arbitrary closed convex cone in <math>\mathcal H</math>. Then, <math>\mathcal K^{\circ\circ}=K.</math> |
Revision as of 17:01, 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: 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.
The extended Farkas' lemma
For any closed convex cone in the Hilbert space , denote by the polar cone of . Let be an arbitrary closed convex cone in . Then,