Moreau's decomposition theorem
(In particular, we can have everywhere in this page.)
Projection on closed convex sets
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
Suppose that Let and be arbitrary. By using the convexity of it follows that Then, by using the definition of the projection, we have
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 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:
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
For definition of convex cone in finite dimension see Convex cones, Wikimization.
For definition of polar cone in finite dimension, see Convex Optimization & Euclidean Distance Geometry.
For applications see Every nonlinear complementarity problem is equivalent to a fixed point problem, Every implicit complementarity problem is equivalent to a fixed point problem, and Projection on isotone projection cone.
- J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962.