Moreau's decomposition theorem
From Wikimization
m (→Every variational inequality defined on a closed convex cone is equivalent to a complementarity problem) 

Line 226:  Line 226:  
Since <math>\mathcal K</math> is a closed convex cone, the nonlinear complementarity problem <math>NCP(f,\mathcal K)</math> is equivalent to the variational inequality <math>VI(f,\mathcal K),</math> which is equivalent to the fixed point problem <math>Fix(P_{\mathcal K}\circ(If)).</math>  Since <math>\mathcal K</math> is a closed convex cone, the nonlinear complementarity problem <math>NCP(f,\mathcal K)</math> is equivalent to the variational inequality <math>VI(f,\mathcal K),</math> which is equivalent to the fixed point problem <math>Fix(P_{\mathcal K}\circ(If)).</math>  
+  
+  == An application to implicit complementarity problems ==  
+  === Implicit complementarity problems ===  
+  
+  Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathbb H,\langle\cdot,\cdot\rangle)</math> and <math>f,g:\mathbb H\to\mathbb H</math> two mappings. 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 '''implicit complementarity problem''' defined by <math>\mathcal K</math>  
+  and the ordered pair of mappings <math>(f,g)\,</math> is the problem  
+  
+  <center>  
+  <math>  
+  ICP(f,g,\mathcal K):\left\{  
+  \begin{array}{l}  
+  Find\,\,\,u\in\mathbb H\,\,\,such\,\,\,that\\  
+  g(u)\in\mathcal K,\,\,\,f(u)\in K^*\,\,\,and\,\,\,\langle g(u),f(u)\rangle=0.  
+  \end{array}  
+  \right.  
+  </math>  
+  </center>  
+  
+  === Every implicit complementarity problem is equivalent to a fixed point problem ===  
+  
+  Let <math>\mathcal K</math> be a closed convex cone in the Hilbert space <math>(\mathbb H,\langle\cdot,\cdot\rangle)</math> and <math>f,g:\mathbb H\to\mathbb H</math> two mappings. Then, the implicit complementarity problem <math>ICP(f,g,\mathcal K)</math> is equivalent to the fixed point problem  
+  <math>Fix(Ig+P_{\mathcal K}\circ(gf)),</math> where <math>I:\mathbb H\to\mathbb H</math> is the identity mapping defined by <math>I(x)=x.\,</math>  
+  
+  === Proof ===  
+  
+  For all <math>u\in\mathbb H</math> denote <math>z=g(u)f(u),\,</math> <math>x=g(u)\,</math> and <math>y=f(u).\,</math> Then,  
+  <math>z=x+y.\,</math>  
+  <br>  
+  <br>  
+  
+  Suppose that <math>u\,</math> is a solution of <math>ICP(f,g,\mathcal K).</math> Then, <math>z=x+y,\,</math> with <math>x\in\mathcal K,</math> <math>y\in\mathcal K^\circ</math> and <math>\langle x,y\rangle=0.</math> Hence, by using  
+  [[Moreau's_decomposition_theorem#Moreau.27s_theorem  Moreau's theorem]],  
+  we get <math>x=P_{\mathcal K}z.</math> Therefore, <math>u\,</math> is a solution of  
+  <math>Fix(Ig+P_{\mathcal K}\circ(gf)).</math>  
+  <br>  
+  <br>  
+  
+  Conversely, suppose that <math>u\,</math> is a solution of <math>Fix(Ig+P_{\mathcal K}\circ(gf)).</math>  
+  Then, <math>x\in\mathcal K</math> and by using  
+  [[Moreau's_decomposition_theorem#Moreau.27s_theorem  Moreau's theorem]]  
+  
+  <center>  
+  <math>z=P_{\mathcal K}(z)+P_{\mathcal K^\circ}(z)=x+P_{\mathcal K^\circ}(z).</math>  
+  </center>  
+  
+  Hence, <math>P_{\mathcal K^\circ}(z)=zx=y,</math>. Thus, <math>y\in\mathcal K^\circ</math>.  
+  [[Moreau's_decomposition_theorem#Moreau.27s_theorem  Moreau's theorem]]  
+  also implies that <math>\langle x,y\rangle=0.</math> In conclusion,  
+  <math>g(u)=x\in\mathcal K,</math> <math>f(u)=y\in\mathcal K^*</math> and <math>\langle x,f(x)\rangle=0.</math> Therefore, <math>u\,</math> is a solution of <math>ICP(f,g,\mathcal K).</math>  
+  
+  === Remark ===  
+  
+  In particular if <math>g=I,</math> we obtain the result of Section  
+  [[Moreau%27s_decomposition_theorem#Every_nonlinear_complementarity_problem_is_equivalent_to_a_fixed_point_problem  3.3 Every nonlinear complementarity problem is equivalent to a fixed point problem]],  
+  but the more general result of Section [[Moreau%27s_decomposition_theorem#Every_implicit_complementarity_problem_is_equivalent_to_a_fixed_point_problem  4.2 Every implicit complementarity problem is equivalent to a fixed point problem]] has no known connection with variational inequalities. 
Revision as of 03:21, 17 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.
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 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: By using the characterization of the projection, we have for all In particular, if then and if then Thus, Denote Then, It remains 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
notes
For definition of convex cone see Convex cone, Wikipedia; in finite dimension see Convex cones, Wikimization.
For definition of polar cone in finite dimension, see more at Dual cone and polar 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.
An application to nonlinear complementarity problems
Fixed point problems
Let be a set and a mapping. The fixed point problem defined by is the problem
Nonlinear complementarity problems
Let be a closed convex cone in the Hilbert space and a mapping. 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 a mapping. Then, the nonlinear complementarity problem is equivalent to the fixed point problem where is the identity mapping defined by
Proof
For all denote and Then,
Suppose that is a solution of Then, with and Hence, by using Moreau's theorem, we get Therefore, is a solution of
Conversely, suppose that is a solution of Then, and by using Moreau's theorem
Hence, . Thus, . Moreau's theorem also implies that In conclusion, and Therefore, is a solution of
An alternative proof without Moreau's theorem
Variational inequalities
Let be a closed convex set in the Hilbert space and a mapping. The variational inequality defined by and is the problem
Every variational inequality is equivalent to a fixed point problem
Let be a closed convex set in the Hilbert space and a mapping. Then the variational inequality is equivalent to the fixed point problem
Proof
is a solution of if and only if By using the characterization of the projection the latter equation is equivalent to
for all But this holds if and only if is a solution of
Remark
The next section shows that the equivalence of variational inequalities and fixed point problems is much stronger than the equivalence of nonlinear complementarity problems and fixed point problems, because each nonlinear complementarity problem is a variational inequality defined on a closed convex cone.
Every variational inequality defined on a closed convex cone is equivalent to a complementarity problem
Let be a closed convex cone in the Hilbert space and a mapping. Then, the nonlinear complementarity problem is equivalent to the variational inequality
Proof
Suppose that is a solution of Then, and Hence,
for all Therefore, is a solution of
Conversely, suppose that is a solution of Then, and
for all Particularly, taking and , respectively, we get Thus, for all or equivalently In conclusion, and Therefore, is a solution of
Concluding the alternative proof
Since is a closed convex cone, the nonlinear complementarity problem is equivalent to the variational inequality which is equivalent to the fixed point problem
An application to implicit complementarity problems
Implicit complementarity problems
Let be a closed convex cone in the Hilbert space and two mappings. Recall that the dual cone of is the closed convex cone where is the polar of The implicit complementarity problem defined by and the ordered pair of mappings is the problem
Every implicit complementarity problem is equivalent to a fixed point problem
Let be a closed convex cone in the Hilbert space and two mappings. Then, the implicit complementarity problem is equivalent to the fixed point problem where is the identity mapping defined by
Proof
For all denote and Then,
Suppose that is a solution of Then, with and Hence, by using
Moreau's theorem,
we get Therefore, is a solution of
Conversely, suppose that is a solution of Then, and by using Moreau's theorem
Hence, . Thus, . Moreau's theorem also implies that In conclusion, and Therefore, is a solution of
Remark
In particular if we obtain the result of Section 3.3 Every nonlinear complementarity problem is equivalent to a fixed point problem, but the more general result of Section 4.2 Every implicit complementarity problem is equivalent to a fixed point problem has no known connection with variational inequalities.