Moreau's decomposition theorem
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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 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. Recall that a closed convex cone in a Hilbert space is a set which is invariant under the addition of vectors and multiplication by nonnegative scalars.
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). 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.