Moreau's decomposition theorem
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| - | <li>1<math>\Rightarrow</math>2: For all <math>p\in K</math> we have <math>\langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0</math>. Then, by the characterization of the projection it follows that <math>x=P_{\mathcal K}z</math>. </li> | + | <li>1<math>\Rightarrow</math>2: For all <math>p\in K</math> we have <math>\langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0</math>. Then, by the characterization of the projection it follows that <math>x=P_{\mathcal K}z</math>. Similarly, for all <math>q\in K^\circ</math>\langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0</math> and thus <math>y=P_{\mathcal K^\circ}z</math>.</li> |
<li>1<math>\Rightarrow</math>2</li> | <li>1<math>\Rightarrow</math>2</li> | ||
</ul> | </ul> | ||
Revision as of 15:36, 10 July 2009
Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.
Let be a closed convex cone in the Hilbert space
and
its polar. For an arbitrary closed convex set
in
, denote by
the projection onto
. For
the following two statements are equivalent:
,
and
and
Proof
Let be an arbitrary closed convex set in
,
and
. Then, it is well known that
if and only if
for all
. We will call this result the characterization of the projection.
- 1
2: For all
we have
. Then, by the characterization of the projection it follows that
\langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0</math> and thus
.
- 1