Complementarity problem
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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 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.