An augmented Lagrangian trust region method for equality constrained optimization

Abstract

In this paper we propose an augmented Lagrangian trust region method for equality constrained optimization. Different from standard augmented Lagrangian methods which minimize the augmented Lagrangian function for fixed Lagrange multiplier and penalty parameter at each iteration, the proposed method tries to minimize its second-order approximation function. We propose a new strategy for adjusting the penalty parameter. With adaptive update of Lagrange multipliers, we prove the global convergence of the proposed method. Numerical results on test problems from the CUTEr collection are also reported.

Keywords:

• equality constraints
• augmented Lagrangian function
• trust region
• Lagrange multiplier
• penalty parameter
• convergence

AMS Subject Classification:

• 90C30
• 65K05

Acknowledgements

The authors would like to thank Prof. M.J.D. Powell for his great suggestions that improved the presentation of the paper, particularly for his providing the proof of Lemma 3.1. The authors are grateful to Prof. Yin Zhang and Prof. Zaiwen Wen for their kind advice on the numerical implementations in this paper and to Prof. Hongchao Zhang for his comments on an earlier version of this paper. They would like to thank two anonymous referees, the associate editor and Prof. Stefan Ulbrich for their detailed and valuable comments and suggestions.

Research of the first author is partially supported by Postdoc Grant 119103S175, UCAS President Grant Y35101AY00 and NSFC grant 11301505. Research of the second author is partially supported by NSFC Grant 11331012.