scholarly journals Global Convergence of Augmented Lagrangian Methods Applied to Optimization Problems with Degenerate Constraints, Including Problems with Complementarity Constraints

2012 ◽  
Vol 22 (4) ◽  
pp. 1579-1606 ◽  
Author(s):  
A. F. Izmailov ◽  
M. V. Solodov ◽  
E. I. Uskov
Keyword(s):  
Global Convergence ◽  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Complementarity Constraints ◽  
Augmented Lagrangian Methods ◽  
Lagrangian Methods ◽  
Degenerate Constraints

Trust Region Augmented Lagrangian Methods With Secant Hessian Updating Applied to Structural Optimization

1996 ◽  
Author(s):  
J. E. Coster ◽  
N. Stander ◽  
J. A. Snyman
Keyword(s):  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Design Method ◽  
Trust Region ◽  
Low Rank ◽  
Truss Structures ◽  
Augmented Lagrangian Methods ◽  
Penalty Term ◽  
Analysis And Design ◽  
Lagrangian Methods

Abstract The problem of determining the optimal sizing design of truss structures is considered. An augmented Lagrangian optimization algorithm which uses a quadratic penalty term is formulated. The implementation uses a first-order Lagrange multiplier update and a strategy for progressively increasing the accuracy with which the bound constrained minimizations are performed. The allowed constraint violation is also progressively decreased but at a slower rate so as to prevent ill-conditioning due to large penalty values. Individual constraint penalties are used and only the penalties of the worst violated constraints are increased. The scheme is globally convergent. The bound constrained minimizations are performed using the SBMIN algorithm where a sophisticated trust-region strategy is employed. The Hessian of the augmented Lagrangian function is approximated using partitioned secant updating. Each function contributing to the Lagrangian is individually approximated by a secant update and the augmented Lagrangian Hessian is formed by appropriate accumulation. The performance of the algorithm is evaluated for a number of different secant updates on standard explicit and truss sizing optimization problems. The results show the formulation to be superior to other implementations of augmented Lagrangian methods reported in the literature and that, under certain conditions, the method approaches the performance of the state-of-the-art SQP and SAM methods. Of the secant updates, the symmetric rank one update is superior to the other updates including the BFGS scheme. It is suggested that the individual function, secant updating employed may be usefully applied in contexts where structural analysis and optimization are performed simultaneously, as in the simultaneous analysis and design method. In such cases the functions are partially separable and the associated Hessians are of low rank.

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