# Augmented Lagrangian

### From Glossary

The Lagrangian augmented by a term that retains
the stationary properties of a solution to the original mathematical
program but alters the Hessian in the subspace
defined by the active set of
constraints. The added term is sometimes called a *penalty
term*, as it decreases the value of the augmented Lagrangian for
off the surface defined by the active constraints. For example,
the penalty term could be proportional to the squared norm of the
active constraints:

where is a parameter (negative for maximization), is the vector of active constraints (at ), and is the usual Lagrangian. In this case, suppose is a Kuhn-Tucker point with multiplier . Then, (since ). Further, the gradient of the penalty term with respect to is at , and the Hessian of the penalty term is simply , where is the matrix whose rows are the gradients of the active constraints. Since is positive semi-definite, the penalty term has the effect of increasing the eigenvalues (decreasing them if maximizing).