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
- 1
2: For all
we have
Then, by the characterization of the projection, it follows that
Similarly, for all
we have
- 2
1: 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 in finite dimension see Convex cones, Wikimization.
For definition of polar cone in finite dimension, see Convex Optimization & Euclidean Distance Geometry.
Applications
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.
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.