Convergence properties of augmented Lagrangian methods for constrained global optimization

Abstract

In this paper, we present new convergence properties of the primal–dual methods based on Rockafellar and Wets's augmented Lagrangian function for inequality constrained global optimization problems. Four different algorithmic strategies are considered to circumvent the boundedness condition of the multipliers in the convergence analysis for basic primal–dual method. We first show that under weaker conditions, the augmented Lagrangian method using safeguarding strategy converges to a global optimal solution of the original problem. The convergence properties of the augmented Lagrangian method using conditional multiplier updating rule is then presented. We also investigate the use of penalty parameter updating criteria and normalization of the multipliers in augmented Lagrangian methods. Finally, we present some preliminary numerical results for the four modified augmented Lagrangian methods.

Keywords:

• constrained global optimization
• augmented Lagrangian methods
• Rockafellar and Wets's function
• convergence to global solution

Acknowledgements

This work was supported by the National Natural Science Foundation of China grants 70671064, 60673177 and 70518001, and the Education Department Foundation of Zhejiang Province grant 20070306.