Moreau's decomposition theorem

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'''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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<ol>
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<li><math>z=x+y</math>, <math>x\in\mathcal K, y\in\mathcal K^\circ</math> and <math>\langle x,y\rangle=0</math></li>
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<li><math>x=P_{\mathcal K}z</math> and <math>y=P_{\mathcal K^\circ}z</math>
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</li>
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</ol>

Revision as of 14:50, 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. Denote by LaTeX: P_{\mathcal K} and LaTeX: P_{\mathcal K^\circ} the projections onto LaTeX: \mathcal K and LaTeX: \mathcal K^\circ, respectively. 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
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