The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems

Abstract

Augmented Lagrangian methods are effective tools for solving large-scale nonlinear programming problems. At each outer iteration, a minimization subproblem with simple constraints, whose objective function depends on updated Lagrange multipliers and penalty parameters, is approximately solved. When the penalty parameter becomes very large, solving the subproblem becomes difficult; therefore, the effectiveness of this approach is associated with the boundedness of the penalty parameters. In this paper, it is proved that under more natural assumptions than the ones employed until now, penalty parameters are bounded. For proving the new boundedness result, the original algorithm has been slightly modified. Numerical consequences of the modifications are discussed and computational experiments are presented.

Keywords:

• nonlinear programming
• augmented Lagrangian methods
• penalty parameters
• numerical experiments

AMS Subject Classifications :

• 90C30
• 49K99
• 65K05

Acknowledgements

We are indebted to two anonymous referees whose comments helped us to improve the paper. This work was supported by PRONEX-CNPq/FAPERJ (E-26/171.164/2003-APQ1), FAPESP (2005/02163-8, 2006/53768-0 and 2008/00062-8), CNPq (480101/2008-6, 303583/2008-8 and 304484/2007-5) and FAPERJ (E-26/102.821/2008).

Notes

The CPLD condition implies that whenever some gradients of active constraints are linearly dependent at a feasible point, and the coefficients corresponding to inequality constraints are non-negative, the same gradients remain linearly dependent in a neighbourhood of the point.