From Wikimization
Sándor Zoltán Németh
(In particular, we can have 
everywhere in this page.)
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, via Moreau's theorem, we get 
Therefore is a solution of 
Conversely, suppose that is a solution of Then 
and via Moreau's theorem
Hence 
thus 
Moreau's theorem also implies that In conclusion,
and 
Therefore is a solution of
Variational inequalities
Let 
be a closed convex set in the Hilbert space and a mapping. The variational inequality defined by and 
is the problem
Remark
The next result is not needed for the alternative proof and it can be skipped. However, it is an important property in its own. It was included for the completeness of the ideas.
Every fixed point problem defined on closed convex set is equivalent to a variational inequality
Let be a closed convex set in the Hilbert space and 
a mapping. Then the fixed point problem 
is equivalent to the variational inequality 
where 
Proof
Suppose that is a solution of . Then,
and thus is a solution of 
Conversely, suppose that is a solution of 
and
let 
Then, 
which is
equivalent to 
Hence, 
; that is, is a solution of .
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
Via characterization of the projection, the latter equation is equivalent to
for all 
But this holds if and only if is a solution
to
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 Choosing 
and 
in particular, we get a system of two inequalities that demands Thus 
for all 
equivalently, 
In conclusion, and Therefore is a solution to
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
Another fixed point problem providing solutions for variational inequalities and complementarity problems
Let be a closed convex set in the Hilbert space 
the identity mapping defined by 
and a mapping. Then, for any solution of the fixed point problem 
the projection 
of onto is a solution of the variational inequality
Proof
Since is a solution of the fixed point problem 
, we have
Thus, 
. Hence, is a solution of the fixed point problem and thus a solution of the variational inequality In particular, if is a closed convex cone, then the variational inequality becomes the complementarity 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
Via Moreau's theorem,
Therefore is a solution of
Conversely, suppose that is a solution of
Then 
and, via
Moreau's theorem,
Hence 
thus 
.
Moreau's theorem
also implies In conclusion,
and 
Therefore is a solution of
Remark
If 
in particular, 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. Using Moreau's theorem is therefore 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 convexity of
we have 
hence
and
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 By
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 
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
Theorem NOPT. 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
Proof
is equivalent to which is equivalent to
An implicit complementarity problem can be associated to an optimization problem
It follows from the definition of an implicit complementarity problem that 
is a solution of 
if and only the minimal value of 
subject to 
and 
is 
at 
Fat nonlinear programming problem
Let 
be a function, 
and
a fat matrix of full rank 
Then the problem
is called fat nonlinear programming problem.
Any solution of a fat nonlinear programming problem is a solution of a nonlinear complementarity problem defined by a polyhedral cone
Let be a differentiable function,
and
a fat matrix of full rank
If 
is a solution of the fat 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 from Theorem NOPT that is a solution of because 
Remark
If is convex, then the converse of the above results also holds.
In other words, 
where the first equivalence has a slightly different meaning than the other ones, but it should be self-explanatory.
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.
Mixed complementarity problems
Let 
and 
be two Hilbert spaces and 
, where 
is an arbitrary closed convex cone in . Let 
, 
and 
. Then, the nonlinear complementarity problem is equivalent to the mixed complementarity problem 
defined by
Proof
Indeed, is a simple equivalent reformulation of the nonlinear complementarity problem , by noting that 
.