Mixed complementarity problems, or box-constrained variational inequalities,
are related to nonlinear systems of equations. They are defined by a
continuously differentiable function,
, and bounds,
and
, on the variables such that
. Given this information,
is a solution to
MCP(F,
, u) if for each
we have at
least one of the following:
Note that when
and
we have a
nonlinear system of equations, and
and
corresponds to the nonlinear complementarity problem [(ref cottle:nonlinear)].
Simple complementarity conditions arise from the first-order optimality
conditions from optimization [(ref karush:minima),(ref kuhn.tucker:nonlinear)].
In the simple bound constrained optimization case, these conditions
correspond to MCP(
,
, u), where
is the objective function. In a one-dimensional setting these conditions
are intuitive. If the solution is at the lower bound, then the function must
be increasing and
. However, if the solution is at the
upper bound, then the function must be decreasing and
.
Finally, if the solution
is strictly between the bounds, we must be at a stationary point and
. Other complementarity problems arise in economics and
engineering [(ref ferris.pang:engineering)], game
theory [(ref nash:equilibrium)], and finance [(ref huang.pang:option)].
Evaluation routines for F and its Jacobian must be supplied prior
to solving the application.
The bounds,
, on the variables must also be
provided.
If no starting point is supplied, a default starting point of all zeros
is used.