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 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 cone (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 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.