# Complementarity problem

### From Wikimization

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Any solution of <math>NOPT(f,\mathcal C)</math> | Any solution of <math>NOPT(f,\mathcal C)</math> | ||

- | [[Complementarity_problem#Any_solution_of_a_nonlinear_optimization_problem_is_a_solution_of_a_variational_inequality | is a solution]] | + | [[Complementarity_problem#Any_solution_of_a_nonlinear_optimization_problem_is_a_solution_of_a_variational_inequality | is a solution of]] <math>VI(F,\mathcal C).</math> |

Conversely, suppose that <math>x\,</math> is a solution of <math>VI(F,\mathcal C).</math> Hence, by using the | Conversely, suppose that <math>x\,</math> is a solution of <math>VI(F,\mathcal C).</math> Hence, by using the |

## Revision as of 05:10, 2 August 2009

Sándor Zoltán Németh

## 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 and is the projection onto

## 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

## 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
*Every nonlinear complementarity problem is equivalent to a fixed point problem*,
but the more general result *Every implicit complementarity problem is equivalent to a fixed point problem* has no known connection with variational inequalities. Therefore, using Moreau's theorem is essential for proving the latter result.

## Nonlinear optimization problems

Let be a closed convex set in the Hilbert space and a function. The
**nonlinear optimization problem** defined by and
is the problem

## Any solution of a nonlinear optimization problem is a solution of a variational inequality

Let be a closed convex set in the Hilbert space and a differentiable function. Then, any solution of the nonlinear optimization problem is a solution of the variational inequality where is the gradient of

## Proof

Let be a solution of and an arbitrary point. Then, by the convexity of we have Hence, and therefore

Therefore, is a solution of

## A convex optimization problem is equivalent to a variational inequality

Let be a closed convex set in the Hilbert space and a differentiable convex function. Then, the nonlinear optimization problem is equivalent to the variational inequality where is the gradient of

## Proof

Any solution of is a solution of

Conversely, suppose that is a solution of Hence, by using the convexity of we have for all Therefore, is a solution of

## Any solution of a nonlinear optimization problem on a closed convex cone is a solution of a nonlinear complementarity problem

Let be a closed convex cone in the Hilbert space and a differentiable function. Then any solution of the nonlinear optimization problem is a solution of the nonlinear complementarity problem where is the gradient of

## Proof

Any solution of is a solution of which is equivalent to

## A convex optimization problem on a closed convex cone is equivalent to a nonlinear complementarity problem

Let be a closed convex cone in the Hilbert space and a differentiable convex function. Then, the nonlinear optimization problem is equivalent to the nonlinear complementarity problem where is the gradient of

## Proof

is equivalent to which is equivalent to

## Non-thin nonlinear programming problem

Let be a function and a matrix of full rank where Then, the problem

is called **non-thin nonlinear programming problem**.

## Any solution of a non-thin nonlinear programming problem is a solution of a nonlinear complementarity problem defined by a polyhedral cone

Let be a differentiable function and a matrix of full rank where If is a solution of the non-thin nonlinear programming problem then is a solution of the nonlinear complementarity problem where is a particular solution of the linear system of equations is the polyhedral cone defined by

and is defined by

## Proof

Let be a solution of Then, it is easy to see that is a solution of where is defined by It follows that is a solution of because

## Remark

If is convex, then the converse of the above results also holds.

We note that there are also many nonlinear programming problems defined by thin matrices (i.e., m>n) which can be reduced to complementarity problems.

Since a very large class of nonlinear programming problems can be reduced to nonlinear complementarity problems, the importance of nonlinear complementarity problems on polyhedral cones is obvious both from theoretical and practical point of view.