Moreau's decomposition theorem

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− Sándor Zoltán Németh

(In particular, we can have \mathbb H=\mathbb R^n everywhere in this page.)

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Projection on closed convex sets

Projection mapping

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

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

Characterization of the projection

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

Proof

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

\|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,

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

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

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

\|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 w\in\mathcal C. Hence, by using the definition of the projection, we get 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.

Theorem (Moreau). Let \mathcal K be a closed convex cone in the Hilbert space (\mathbb H,\langle\cdot,\cdot\rangle) and \mathcal K^\circ its polar cone; that is, the closed convex cone defined by \mathcal K^\circ=\{a\in\mathbb H\mid\langle a,b\rangle\leq0,\,\forall b\in\mathcal K\}.

For x,y,z\in\mathbb H the following statements are equivalent:

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

Proof of Moreau's theorem

  • 1\Rightarrow2: For all p\in K we have

    \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 x=P_{\mathcal K}z. Similarly, for all q\in K^\circ we have

    \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0

    and thus y=P_{\mathcal K^\circ}z.
  • 2\Rightarrow1: By using the characterization of the projection, we have \langle z-x,p-x\rangle\leq0, for all p\in\mathcal K. In particular, if p=0,\, then \langle z-x,x\rangle\geq0 and if p=2x,\, then \langle z-x,x\rangle\leq0. Thus, \langle z-x,x\rangle=0. Denote u=z-x.\, Then \langle x,u\rangle=0. It remains to show that u=y.\, First, we prove that u\in\mathcal K^\circ. For this we have to show that \langle u,p\rangle\leq0, for all p\in\mathcal K. By using the characterization of the projection, we have

    \langle u,p\rangle=\langle u,p-x\rangle=\langle z-x,p-x\rangle\leq0,

    for all p\in\mathcal K. Thus, u\in\mathcal K^\circ. We also have

    \langle z-u,q-u\rangle=\langle x,q-u\rangle=\langle x,q\rangle\leq0,

    for all q\in K^\circ, because x\in K. By using again the characterization of the projection, it follows that u=y.\,

notes

For definition of convex cone in finite dimension see Convex cones, Wikimization.

For definition of polar cone in finite dimension, see Convex Optimization & Euclidean Distance Geometry.

\mathcal K^{\circ\circ}=K see Extended Farkas' lemma.

Applications

For applications see Every nonlinear complementarity problem is equivalent to a fixed point problem, Every implicit complementarity problem is equivalent to a fixed point problem, and Projection on isotone projection cone.

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.
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