Convergence Properties of an Augmented Lagrangian
Algorithm for Optimization with a Combination of
General Equality and Linear Constraints
A. R. Conn, Nick Gould, A. Sartenaer and Ph.L. Toint
We consider the global and local convergence
properties of a class of augmented Lagrangian methods
for solving nonlinear programming problems. In these
methods, linear and more general constraints are
handled in different ways. The general constraints
are combined with the objective function in an
augmented Lagrangian. The iteration consists of
solving a sequence of subproblems; in each subproblem
the augmented Lagrangian is approximately minimized
in the region defined by the linear constraints. A
subproblem is terminated as soon as a stopping
condition is satisfied. The stopping rules that we
consider here encompass practical tests used in
several existing packages for linearly constrained
optimization. Our algorithm also allows different
penalty parameters to be associated with disjoint
subsets of the general constraints. In this paper,
we analyze the convergence of the sequence of
iterates generated by such an algorithm and prove
global and fast linear convergence as well as showing
that potentially troublesome penalty parameters
remain bounded away from zero.