Moreau's decomposition theorem

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Revision as of 04:51, 11 July 2009

1 Projection mapping
2 Characterization of the projection
3 Proof
4 Moreau's theorem
5 Proof of Moreau's theorem
6 References

Projection mapping

Let $LaTeX: (\mathcal H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and $LaTeX: \mathcal C$ a closed convex set in $LaTeX: \mathcal H$ . The projection mapping $LaTeX: P_{\mathcal C}$ onto $LaTeX: \mathcal C$ is the mapping $LaTeX: P_{\mathcal C}:\mathcal H\to\mathcal H$ defined by $LaTeX: P_{\mathcal C}(x)\in\mathcal C$ and

$LaTeX: \|x-P_{\mathcal C}(x)\|=\min\{\|x-y\|\mid y\in\mathcal C\}.$

Characterization of the projection

Let $LaTeX: (\mathcal H,\langle\cdot,\cdot\rangle)$ be a Hilbert space, $LaTeX: \mathcal C$ a closed convex set in $LaTeX: \mathcal H,\,u\in\mathcal H$ and $LaTeX: v\in\mathcal C$ . Then, $LaTeX: v=P_{\mathcal C}(u)$ if and only if $LaTeX: \langle u-v,w-v\rangle\leq0$ for all $LaTeX: w\in\mathcal C$ .

Proof

Suppose that $LaTeX: v=P_{\mathcal C}u$ and let $LaTeX: w\in\mathcal C$ be arbitrary. By using the convexity of $LaTeX: \mathcal C$ , it follows that $LaTeX: (1-t)v+tw\in\mathcal C$ , for all $LaTeX: t\in (0,1)$ . Then, by using the definition of the projection, we have

$LaTeX: \|u-v\|^2\leq\|u-[(1-t)v+tw]\|^2=\|u-v-t(w-v)\|^2=\|u-v\|^2-2t\langle u-v,w-v\rangle+t^2\|w-v\|^2$ .

Hence,

$LaTeX: \langle u-v,w-v\rangle\leq\frac t2\|w-v\|^2.$

By tending with $LaTeX: t$ to $LaTeX: 0$ , we get $LaTeX: \langle u-v,w-v\rangle\leq0$ .

Conversely, suppose that $LaTeX: \langle u-v,w-v\rangle\leq0,$ for all $LaTeX: w\in\mathcal C$ . Then,

$LaTeX: \|u-w\|^2=\|u-v-(w-v)\|^2=\|u-v\|^2-2\langle u-v,w-v\rangle+\|w-v\|^2\geq \|u-v\|^2,$

for all $LaTeX: w\in\mathcal C$ . Hence, by using the definition of the projection, we get $LaTeX: v=P_{\mathcal C}u$ .

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 (see more at Convex cone (Wikipedia) or for finite dimension at Convex cone (Wikimization)).

Theorem (Moreau) 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 cone; that is, the closed convex cone defined by $LaTeX: K^\circ=\{a\in\mathcal H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}$ (for finite dimension see more at Dual cone and polar cone; see also Extended Farkas' lemma). For $LaTeX: x,y,z\in\mathcal H$ the following statements are equivalent:

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

Proof of Moreau's theorem

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

Then, by the characterization of the projection, it follows that $LaTeX: x=P_{\mathcal K}z$ . Similarly, for all $LaTeX: q\in K^\circ$ we have

$LaTeX: \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0$
and thus $LaTeX: y=P_{\mathcal K^\circ}z$ .
2 $LaTeX: \Rightarrow$ 1: Let $LaTeX: x=P_{\mathcal K}z$ . By the characterization of the projection we have $LaTeX: \langle z-x,p-x\rangle\leq0,$ for all $LaTeX: p\in\mathcal K$ . In particular, if $LaTeX: p=0,$ then $LaTeX: \langle z-x,x\rangle\geq0$ and if $LaTeX: p=2x,$ then $LaTeX: \langle z-x,x\rangle\leq0$ . Thus, $LaTeX: \langle z-x,x\rangle=0$ . Denote $LaTeX: y=z-x$ . Then, $LaTeX: \langle x,y\rangle=0$ . It remained to show that $LaTeX: y=P_{\mathcal K^\circ}z$ . First, we prove that $LaTeX: y\in\mathcal K^\circ$ . For this we have to show that $LaTeX: \langle y,p\rangle\leq0$ , for all $LaTeX: p\in\mathcal K$ . By using the characterization of the projection, we have
$LaTeX: \langle y,p\rangle=\langle y,p-x\rangle=\langle z-x,p-x\rangle\leq0,$

for all $LaTeX: p\in\mathcal K$ . Thus, $LaTeX: y\in\mathcal K^\circ$ . We also have

$LaTeX: \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0,$

for all $LaTeX: q\in K^\circ$ , because $LaTeX: x\in K$ . By using again the characterization of the projection, it follows that $LaTeX: y=P_{\mathcal K^\circ}z$ .

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.

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Moreau%27s_decomposition_theorem"

@@ Line 44: / Line 44: @@
 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 (see more at [http://en.wikipedia.org/wiki/Convex_cone Convex cone (Wikipedia)] or for finite dimension at [http://www.convexoptimization.com/wikimization/index.php/Convex_cones Convex cone (Wikimization)]).
+under the addition of vectors and multiplication of vectors by positive scalars (see more at [http://en.wikipedia.org/wiki/Convex_cone Convex cone (Wikipedia)] or for finite dimension at [[Convex cones | Convex cone (Wikimization)]]).
-'''Theorem (Moreau)''' 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 cone'''; that is, the closed convex cone defined by <math>K^\circ=\{a\in\mathcal H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}</math> (for finite dimension see more at [http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone Dual cone and polar cone]; see also [http://www.convexoptimization.com/wikimization/index.php/Farkas%27_lemma#Extended_Farkas.27_lemma Extended Farkas' lemma]). For <math>x,y,z\in\mathcal H</math> the following statements are equivalent:
+'''Theorem (Moreau)''' 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 cone'''; that is, the closed convex cone defined by <math>K^\circ=\{a\in\mathcal H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}</math> (for finite dimension see more at [http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone Dual cone and polar cone]; see also [[Farkas%27_lemma#Extended_Farkas.27_lemma | Extended Farkas' lemma]]). For <math>x,y,z\in\mathcal H</math> the following statements are equivalent:
 <ol>

Moreau's decomposition theorem

From Wikimization

Revision as of 04:51, 11 July 2009

Contents

Projection mapping

Characterization of the projection

Proof

Moreau's theorem

Proof of Moreau's theorem

References

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