Moreau's decomposition theorem
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m (→Projection on closed convex sets) 
(→Moreau's theorem) 

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* J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962.  * J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962.  
+  
+  == An application to nonlinear complementarity problems ==  
+  
+  === Fixed point problems ===  
+  
+  Let <math>\mathcal A</math> be a set and <math>F:\mathcal A\to\mathcal A.</math> The '''fixed point problem''' defined by <math>F\,</math> is the problem  
+  
+  <center>  
+  <math>  
+  Fix(F):\left\{  
+  \begin{array}{l}  
+  Find\,\,\,x\in\mathcal A\,\,\,such\,\,\,that\\  
+  F(x)=x.  
+  \end{array}  
+  \right.  
+  </math>  
+  </center>  
+  
+  === Nonlinear complementarity problems ===  
+  
+  Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>f:\mathcal H\to\mathcal H.</math> Recall that the dual cone of <math>\mathcal K</math> is the closed convex cone <math>\mathcal K^*=\mathcal K^\circ,</math> where <math>\mathcal K^\circ</math> is the [[Moreau's_decomposition_theorem#Moreau.27s_theorem polar]] of <math>\mathcal K.</math> The '''nonlinear complementarity problem''' defined by <math>\mathcal K</math> and <math>f\,</math> is the problem  
+  
+  <center>  
+  <math>  
+  NCP(f,\mathcal K):\left\{  
+  \begin{array}{l}  
+  Find\,\,\,x\in\mathcal K\,\,\,such\,\,\,that\\  
+  f(x)\in\mathcal K^*\,\,\,and\,\,\,\langle x,f(x)\rangle=0.  
+  \end{array}  
+  \right.  
+  </math>  
+  </center>  
+  
+  === Every nonlinear complementarity problem is equivalent to a fixed point problem ===  
+  
+  Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathcal H,\langle\cdot,\cdot\rangle)</math> and <math>f:\mathcal H\to\mathcal H.</math> Then, the nonlinear complementarity problem <math>NCP(f,\mathcal K)</math> is equivalent to the fixed point problem  
+  <math>Fix(P_{\mathcal K}\circ(If)),</math> where <math>I:\mathcal H\to\mathcal H</math> is the identity mapping defined by <math>I(x)=x.\,</math>  
+  
+  === Proof ===  
+  
+  === An alternative proof without Moreau's theorem ===  
+  
+  ==== Variational inequalities ====  
+  
+  ==== Every variational inequality is equivalent to a fixed point problem ====  
+  
+  ==== Every variational inequality defined on a closed convex cone is equivalent to a complementarity problem ==== 
Revision as of 06:11, 12 July 2009
Contents

Projection on closed convex sets
Projection mapping
Let be a Hilbert space and a closed convex set in The projection mapping onto is the mapping defined by and
Characterization of the projection
Let be a Hilbert space, a closed convex set in and Then, if and only if for all
Proof
Suppose that Let and be arbitrary. By using the convexity of it follows that Then, by using the definition of the projection, we have
Hence,
By tending with to we get
Conversely, suppose that for all Then,
for all Hence, by using the definition of the projection, we get
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 cones, Wikimization).
Theorem (Moreau) Let be a closed convex cone in the Hilbert space and its polar cone; that is, the closed convex cone defined by (for finite dimension see more at Dual cone and polar cone; see also Extended Farkas' lemma). For the following statements are equivalent:
 and
 and
Proof of Moreau's theorem
 12: For all we have
.
Then, by the characterization of the projection, it follows that . Similarly, for all we have
 21: Let . By using the characterization of the projection, we have for all . In particular, if then and if then . Thus, . Denote . Then, . It remained to show that . First, we prove that . For this we have to show that , for
all . By using the characterization of the projection, we have
for all . Thus, . We also have
for all , because . By using again the characterization of the projection, it follows that .
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.
An application to nonlinear complementarity problems
Fixed point problems
Let be a set and The fixed point problem defined by is the problem
Nonlinear complementarity problems
Let be a closed convex cone in the Hilbert space and Recall that the dual cone of is the closed convex cone where is the polar of The nonlinear complementarity problem defined by and is the problem
Every nonlinear complementarity problem is equivalent to a fixed point problem
Let be a closed convex cone in the Hilbert space and Then, the nonlinear complementarity problem is equivalent to the fixed point problem where is the identity mapping defined by