Moreau's decomposition theorem

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'''Moreau's theorem''' is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.
'''Moreau's theorem''' is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.
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Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>\mathcal K^\circ</math> its polar. Denote by <math>P_{\mathcal K}</math> and <math>P_{\mathcal K^\circ}</math> the projections onto <math>\mathcal K</math> and <math>\mathcal K^\circ</math>, respectively. For <math>x,y,z\in\mathcal H</math> the following two statements are equivalent:
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Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>\mathcal K^\circ</math> its polar. For an arbitrary closed convex set <math>\mathcal C</math> in <math>\mathcal H</math>, denote by <math>P_{\mathcal C}</math> the projection onto <math>\mathcal C</math>. For <math>x,y,z\in\mathcal H</math> the following two statements are equivalent:
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== Proof ==
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Let <math>\mathcal C</math> be an arbitrary closed convex set in <math>\mathcal H</math>, <math>u\in\mathcal H</math> and <math>v\in\mathcal C</math>. Then, it is well known that <math>v=P_{\mathcal C}u</math> if and only if <math>\langle u-v,w-v\rangle\leq0</math> for all <math>w\in\mathcal C</math>. We will call this result the '''''characterization of the projection'''''.
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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></li>
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<li>1<math>\Rightarrow</math>2</li>
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</ul>

Revision as of 15:25, 10 July 2009

Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.

Let LaTeX: \mathcal K be a closed convex cone in the Hilbert space LaTeX: (\mathcal H,\langle\cdot,\cdot\rangle) and LaTeX: \mathcal K^\circ its polar. For an arbitrary closed convex set LaTeX: \mathcal C in LaTeX: \mathcal H, denote by LaTeX: P_{\mathcal C} the projection onto LaTeX: \mathcal C. For LaTeX: x,y,z\in\mathcal H the following two statements are equivalent:

  1. LaTeX: z=x+y, LaTeX: x\in\mathcal K, y\in\mathcal K^\circ and LaTeX: \langle x,y\rangle=0
  2. LaTeX: x=P_{\mathcal K}z and LaTeX: y=P_{\mathcal K^\circ}z

Proof

Let LaTeX: \mathcal C be an arbitrary closed convex set in LaTeX: \mathcal H, LaTeX: u\in\mathcal H and LaTeX: v\in\mathcal C. Then, it is well known that LaTeX: v=P_{\mathcal C}u if and only if LaTeX: \langle u-v,w-v\rangle\leq0 for all LaTeX: w\in\mathcal C. We will call this result the characterization of the projection.

  • 1LaTeX: \Rightarrow2: For all LaTeX: p\in K we have LaTeX: \langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0
  • 1LaTeX: \Rightarrow2
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