# Moreau's decomposition theorem

(Difference between revisions)
 Revision as of 14:50, 10 July 2009 (edit)← Previous diff Revision as of 15:25, 10 July 2009 (edit) (undo)Next diff → Line 1: Line 1: '''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. - Let $\mathcal K$ be a closed convex cone in the Hilbert space $(\mathcal H,\langle\cdot,\cdot\rangle)$ and $\mathcal K^\circ$ its polar. Denote by $P_{\mathcal K}$ and $P_{\mathcal K^\circ}$ the projections onto $\mathcal K$ and $\mathcal K^\circ$, respectively. For $x,y,z\in\mathcal H$ the following two statements are equivalent: + Let $\mathcal K$ be a closed convex cone in the Hilbert space $(\mathcal H,\langle\cdot,\cdot\rangle)$ and $\mathcal K^\circ$ its polar. For an arbitrary closed convex set $\mathcal C$ in $\mathcal H$, denote by $P_{\mathcal C}$ the projection onto $\mathcal C$. For $x,y,z\in\mathcal H$ the following two statements are equivalent:
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+ + == Proof == + + Let $\mathcal C$ be an arbitrary closed convex set in $\mathcal H$, $u\in\mathcal H$ and $v\in\mathcal C$. Then, it is well known that $v=P_{\mathcal C}u$ if and only if $\langle u-v,w-v\rangle\leq0$ for all $w\in\mathcal C$. We will call this result the '''''characterization of the projection'''''. + +
+
• 1$\Rightarrow$2: For all $p\in K$ we have $\langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0$
• +
• 1$\Rightarrow$2
• +

## 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.

• 1$LaTeX: \Rightarrow$2: For all $LaTeX: p\in K$ we have $LaTeX: \langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0$
• 1$LaTeX: \Rightarrow$2