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Complementarity problem

2012-04-20T13:33:16Z

←Older revision		Revision as of 13:33, 20 April 2012
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	==== Concluding the alternative proof ====		==== Concluding the alternative proof ====
	Since <math>\mathcal{K}</math> is a closed convex cone, the [[Complementarity_problem#Nonlinear_complementarity_problems \| nonlinear complementarity problem]] <math>NCP(F,\mathcal{K})</math> is equivalent to the [[Complementarity_problem#Variational_inequalities \| variational inequality]] <math>VI(F,\mathcal{K})</math> which [[Complementarity_problem#Every_variational_inequality_is_equivalent_to_a_fixed_point_problem \| is equivalent to]] the [[Complementarity_problem#Fixed_point_problems \| fixed point problem]] <math>Fix(P_{\mathcal{K}}\circ(I-F)).</math>		Since <math>\mathcal{K}</math> is a closed convex cone, the [[Complementarity_problem#Nonlinear_complementarity_problems \| nonlinear complementarity problem]] <math>NCP(F,\mathcal{K})</math> is equivalent to the [[Complementarity_problem#Variational_inequalities \| variational inequality]] <math>VI(F,\mathcal{K})</math> which [[Complementarity_problem#Every_variational_inequality_is_equivalent_to_a_fixed_point_problem \| is equivalent to]] the [[Complementarity_problem#Fixed_point_problems \| fixed point problem]] <math>Fix(P_{\mathcal{K}}\circ(I-F)).</math>
		+
		+	== Another fixed point problem providing solutions for variational inequalities and complementarity problems ==
		+
		+	Let <math>\mathcal{C}</math> be a closed convex set in the Hilbert space <math>(\mathbb{H},\langle\cdot,\cdot\rangle),</math>
		+	<math>I:\,\mathbb{H}\to\mathbb{H}</math> the identity mapping defined by <math>I(x)=x</math> and <math>F:\,\mathbb{H}\to\mathbb{H}</math> a mapping. Then, for any solution <math>x\,</math> of the fixed point problem <math>Fix({(I-F)\circ P_{\mathcal{C}}}),</math> the projection <math>P_{\mathcal{C}}(x)</math> of <math>x\,</math> onto <math>\mathcal{C}</math>is a solution of the variational inequality <math>VI(F,\mathcal{C}).</math>
		+
		+	=== Proof ===
		+	Since <math>x\,</math> is a solution of the fixed point problem <math>Fix({(I-F)\circ P_{\mathcal{C}}})</math>, we have
		+	<math>P_{\mathcal{C}}(x)-F(P_{\mathcal{C}}(x))=x.</math> Thus, <math>P_{\mathcal{C}}(P_{\mathcal{C}}(x)-F(P_{\mathcal{C}}(x)))=P_{\mathcal{C}}(x)</math>. Hence, <math>P_{\mathcal{C}}(x)</math> is a solution of the fixed point problem <math>Fix(P_{\mathcal{C}}\circ(I-F))</math> and [[Complementarity_problem#Every_variational_inequality_is_equivalent_to_a_fixed_point_problem \| thus]] a solution of the variational inequality <math>VI(F,\mathcal{C}).</math> In particular, if <math>\mathcal{C}</math> is a closed convex cone, then the variational inequality <math>VI(F,\mathcal{C})</math> [[Complementarity_problem#Every_variational_inequality_defined_on_a_closed_convex_cone_is_equivalent_to_a_complementarity_problem \| becomes]] the complementarity problem <math>NCP(F,\mathcal{C}).</math>

	== Implicit complementarity problems ==		== Implicit complementarity problems ==
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	<center>		<center>
	<math>		<math>
-	ICP(F,G,\mathcal K):\,\left\{	+	ICP(F,G,\mathcal{K}):\,\left\{
	\begin{array}{l}		\begin{array}{l}
	\textrm{Find }u\in\mathbb{H}\textrm{ such that}\\\\\\		\textrm{Find }u\in\mathbb{H}\textrm{ such that}\\\\\\
-	G(u)\in\mathcal K,\,\,\,F(u)\in K^*,\,\,\,\langle G(u),F(u)\rangle=0.	+	G(u)\in\mathcal{K},\,\,\,F(u)\in\mathcal{K}^*,\,\,\,\langle G(u),F(u)\rangle=0.
	\end{array}		\end{array}
	\right.		\right.
	</math>		</math>
	</center>		</center>
-
-	== Another fixed point problem providing solutions for variational inequalities and complementarity problems ==
-
-	Let <math>\mathcal{C}</math> be a closed convex set in the Hilbert space <math>(\mathbb{H},\langle\cdot,\cdot\rangle),</math>
-	<math>I:\,\mathbb{H}\to\mathbb{H}</math> the identity mapping defined by <math>I(x)=x</math> and <math>F:\,\mathbb{H}\to\mathbb{H}</math> a mapping. Then, for any solution <math>x\,</math> of the fixed point problem <math>Fix({(I-F)\circ P_{\mathcal{C}}}),</math> the projection <math>P_{\mathcal{C}}(x)</math> of <math>x\,</math> onto <math>\mathcal{C}</math>is a solution of the variational inequality <math>VI(F,\mathcal{C}).</math>
-
-	=== Proof ===
-	Since <math>x\,</math> is a solution of the fixed point problem <math>Fix({(I-F)\circ P_{\mathcal{C}}})</math>, we have
-	<math>P_{\mathcal{C}}(x)-F(P_{\mathcal{C}}(x))=x.</math> Thus, <math>P_{\mathcal{C}}(P_{\mathcal{C}}(x)-F(P_{\mathcal{C}}(x)))=P_{\mathcal{C}}(x)</math>. Hence, <math>P_{\mathcal{C}}(x)</math> is a solution of the fixed point problem <math>Fix(P_{\mathcal{C}}\circ(I-F))</math> and [[Complementarity_problem#Every_variational_inequality_is_equivalent_to_a_fixed_point_problem \| thus]] a solution of the variational inequality <math>VI(F,\mathcal{C}).</math> In particular, if <math>\mathcal{C}</math> is a closed convex cone, then the variational inequality <math>VI(F,\mathcal{C})</math> [[Complementarity_problem#Every_variational_inequality_defined_on_a_closed_convex_cone_is_equivalent_to_a_complementarity_problem \| becomes]] the complementarity problem <math>NCP(F,\mathcal{C}).</math>

	== Every implicit complementarity problem is equivalent to a fixed point problem ==		== Every implicit complementarity problem is equivalent to a fixed point problem ==
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	=== Proof ===		=== Proof ===
	<math>NOPT(f,\mathcal K)</math> [[Complementarity_problem#A_convex_optimization_problem_is_equivalent_to_a_variational_inequality \| is equivalent to]] <math>VI(\nabla f,\mathcal K)</math> which [[Complementarity_problem#Every_variational_inequality_defined_on_a_closed_convex_cone_is_equivalent_to_a_complementarity_problem\| is equivalent to]] <math>NCP(\nabla f,\mathcal K).</math>		<math>NOPT(f,\mathcal K)</math> [[Complementarity_problem#A_convex_optimization_problem_is_equivalent_to_a_variational_inequality \| is equivalent to]] <math>VI(\nabla f,\mathcal K)</math> which [[Complementarity_problem#Every_variational_inequality_defined_on_a_closed_convex_cone_is_equivalent_to_a_complementarity_problem\| is equivalent to]] <math>NCP(\nabla f,\mathcal K).</math>
		+
		+	== An implicit complementarity problem can be associated to an optimization problem ==
		+
		+	It follows from the definition of an [[Complementarity_problem#Implicit_complementarity_problems \| implicit complementarity problem]] that <math>y</math> is a solution of <math>ICP(F,G,\mathcal{K})</math> if and only the minimal value of <math>\langle G(x),F(x)\rangle</math> subject to <math>G(x)\in\mathcal{K}</math> and <math>F(x)\in\mathcal{K}^*</math> is <math>0</math> at <math>y.</math>

	== Fat nonlinear programming problem ==		== Fat nonlinear programming problem ==

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Complementarity problem