Numerical Methods for Variational Inequalities and Optimal Control

Weak formulations of partial differential equations can, as shown in Chapter 3, be written as variational equations. They also give necessary and sufficient optimality conditions for the minimization of convex functionals on some linear subspace or linear manifold in an appropriate function space. In models that lead to variational problems, the constraints that appear have often a structure that does not produce problems posed on subspaces. Then optimality conditions do not lead automatically to variational equations, but may instead be formulated as variational inequalities. In the present section, several important properties of variational inequalities are collected and we sketch their connection to the problems of convex analysis.