Improving the Performance of the Augmented Lagrangian Coordination: Decomposition Variants and Dual Residuals

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Meng Xu ◽  
Georges Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Test Problem ◽  
Augmented Lagrangian ◽  
Analytical Target Cascading ◽  
Advantages And Disadvantages ◽  
Numerical Tests ◽  
Target Cascading ◽  
Augmented Lagrangian Coordination

The augmented Lagrangian coordination (ALC), as an effective coordination method for decomposition-based optimization, offers significant flexibility by providing different variants when solving nonhierarchically decomposed problems. In this paper, these ALC variants are analyzed with respect to the number of levels and multipliers, and the resulting advantages and disadvantages are explored through numerical tests. The efficiency, accuracy, and parallelism of three ALC variants (distributed ALC, centralized ALC, and analytical target cascading (ATC) extended by ALC) are discussed and compared. Furthermore, the dual residual theory for the centralized ALC is extended to the distributed ALC, and a new flexible nonmonotone weight update is proposed and tested. Numerical tests show that the proposed update effectively improves the accuracy and robustness of the distributed ALC on a benchmark engineering test problem.

Cutting Plane Methods for Analytical Target Cascading With Augmented Lagrangian Coordination

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Wenshan Wang ◽  
Vincent Y. Blouin ◽  
Melissa K. Gardenghi ◽  
Georges M. Fadel ◽  
Margaret M. Wiecek ◽  
...  
Keyword(s):  
Large Scale ◽  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Analytical Target Cascading ◽  
Cutting Plane Methods ◽  
Subgradient Optimization ◽  
Decomposition Approach ◽  
Target Cascading ◽  
Augmented Lagrangian Coordination

Analytical target cascading (ATC), a hierarchical, multilevel, multidisciplinary coordination method, has proven to be an effective decomposition approach for large-scale engineering optimization problems. In recent years, augmented Lagrangian relaxation methods have received renewed interest as dual update methods for solving ATC decomposed problems. These problems can be solved using the subgradient optimization algorithm, the application of which includes three schemes for updating dual variables. To address the convergence efficiency disadvantages of the existing dual update schemes, this paper investigates two new schemes, the linear and the proximal cutting plane methods, which are implemented in conjunction with augmented Lagrangian coordination for ATC-decomposed problems. Three nonconvex nonlinear example problems are used to show that these two cutting plane methods can significantly reduce the number of iterations and the number of function evaluations when compared to the traditional subgradient update methods. In addition, these methods are also compared to the method of multipliers and its variants, showing similar performance.

Dual Residual for Centralized Augmented Lagrangian Coordination Based on Optimality Conditions

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Meng Xu ◽  
Georges Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Optimality Conditions ◽  
Parallel Computation ◽  
Augmented Lagrangian ◽  
Numerical Tests ◽  
Wide Range ◽  
Initial Setting ◽  
Iteration Number ◽  
Update Strategy ◽  
Solution Accuracy ◽  
Augmented Lagrangian Coordination

Centralized augmented Lagrangian coordination (ALC) has drawn much attention due to its parallel computation capability, efficiency, and flexibility. The initial setting and update strategy of the penalty weights in this method are critical to its performance. The traditional weight update strategy always increases the weights and research shows that inappropriate initial weights may cause optimization failure. Making use of the Karush–Kuhn–Tucker (KKT) optimality conditions for the all-in-one (AIO) and decomposed problems, the terms “primal residual” and “dual residual” are introduced into the centralized ALC, and a new update strategy considering both residuals and thus guaranteeing the unmet optimality condition in the traditional update is introduced. Numerical tests show a decrease in the iteration number and significant improvements in solution accuracy with both calculated and fine-tuned initial weights using the new update. Additionally, the proposed approach is capable to start from a wide range of possible weights and achieve optimality, and therefore brings robustness to the centralized ALC.

A Cutting Plane Method for Analytical Target Cascading With Augmented Lagrangian Coordination

2010 ◽  
Author(s):  
Wenshan Wang ◽  
Vincent Y. Blouin ◽  
Melissa Gardenghi ◽  
Margaret M. Wiecek ◽  
Georges M. Fadel ◽  
...  
Keyword(s):  
Large Scale ◽  
Augmented Lagrangian ◽  
Cutting Plane ◽  
Computational Effort ◽  
Subgradient Optimization ◽  
Augmented Lagrangian Methods ◽  
Design Problems ◽  
Target Cascading ◽  
Decomposition And Coordination ◽  
Augmented Lagrangian Coordination

Various decomposition and coordination methodologies for solving large-scale system design problems have been developed and studied during the past few decades. However, there is generally no guarantee that they will converge to the expected optimum design under general assumptions. Those with proven convergence often have restricted hypotheses or a prohibitive cost related to the required computational effort. Therefore there is still a need for improved, mathematically grounded, decomposition and coordination techniques that will achieve convergence while remaining robust, flexible and easy to implement. In recent years, classical Lagrangian and augmented Lagrangian methods have received renewed interest when applied to decomposed design problems. Some methods are implemented using a subgradient optimization algorithm whose performance is highly dependent on the type of dual update of the iterative process. This paper reports on the implementation of a cutting plane approach in conjunction with Lagrangian coordination and the comparison of its performance with other subgradient update methods. The method is demonstrated on design problems that are decomposable according to the analytic target cascading (ATC) scheme.

A Nonhierarchical Formulation of Analytical Target Cascading

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
S. Tosserams ◽  
M. Kokkolaras ◽  
L. F. P. Etman ◽  
J. E. Rooda
Keyword(s):  
System Design ◽  
Problem Formulation ◽  
Optimal System ◽  
System Level ◽  
Master Problem ◽  
Analytical Target Cascading ◽  
Target Cascading ◽  
Optimal System Design ◽  
New Formulation ◽  
Augmented Lagrangian Coordination

Analytical target cascading (ATC) is a method developed originally for translating system-level design targets to design specifications for the components that comprise the system. ATC has been shown to be useful for coordinating decomposition-based optimal system design. The traditional ATC formulation uses hierarchical problem decompositions, in which coordination is performed by communicating target and response values between parents and children. The hierarchical formulation may not be suitable for general multidisciplinary design optimization (MDO) problems. This paper presents a new ATC formulation that allows nonhierarchical target-response coupling between subproblems and introduces system-wide functions that depend on variables of two or more subproblems. Options to parallelize the subproblem optimizations are also provided, including a new bilevel coordination strategy that uses a master problem formulation. The new formulation increases the applicability of the ATC to both decomposition-based optimal system design and MDO. Moreover, it belongs to the class of augmented Lagrangian coordination methods, having thus convergence properties under standard convexity and continuity assumptions. A supersonic business jet design problem is used to demonstrate the flexibility and effectiveness of the presented formulation.

scholarly journals An Enhanced Analytical Target Cascading and Kriging Model Combined Approach for Multidisciplinary Design Optimization

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Ping Jiang ◽  
Jianzhuang Wang ◽  
Qi Zhou ◽  
Xiaolin Zhang
Keyword(s):  
Design Optimization ◽  
Multidisciplinary Design Optimization ◽  
Kriging Model ◽  
Multidisciplinary Design ◽  
High Approximation ◽  
Engineering Systems ◽  
Analytical Target Cascading ◽  
Combined Approach ◽  
Coordination Strategy ◽  
Target Cascading

Multidisciplinary design optimization (MDO) has been applied widely in the design of complex engineering systems. To ease MDO problems, analytical target cascading (ATC) organizes MDO process into multilevels according to the components of engineering systems, which provides a promising way to deal with MDO problems. ATC adopts a coordination strategy to coordinate the couplings between two adjacent levels in the design optimization process; however, existing coordination strategies in ATC face the obstacles of complicated coordination process and heavy computation cost. In order to conquer this problem, a quadratic exterior penalty function (QEPF) based ATC (QEPF-ATC) approach is proposed, where QEPF is adopted as the coordination strategy. Moreover, approximate models are adopted widely to replace the expensive simulation models in MDO; a QEPF-ATC and Kriging model combined approach is further proposed to deal with MDO problems, owing to the comprehensive performance, high approximation accuracy, and robustness of Kriging model. Finally, the geometric programming and reducer design cases are given to validate the applicability and efficiency of the proposed approach.

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