Moreau's decomposition theorem
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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.
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 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: By using the characterization of the projection, we have for all In particular, if then and if then Thus, Denote Then, It remains 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
notes
For definition of convex cone see Convex cone, Wikipedia; in finite dimension see Convex cones, Wikimization.
For definition of polar cone in finite dimension, see more at Dual cone and polar cone.
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.