Variational Inequality

Let be a nonempty closed convex set in a Hilbert space , let be an operator, and let . A variational inequality 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.