A variational inequality on a nonempty closed convex set 
in a Hilbert
space 
,
for an operator 
and 
, asks for 
such that
 |
for every 
.
When 
is the gradient of a convex functional,
this condition characterizes an associated constrained minimization problem. Obstacle
problems for partial differential equations
are an important class of variational inequalities. In particular, the value of an
American option can be characterized by a variational
inequality whose contact set is the exercise region and whose interface is the boundary in a free
boundary problem.
