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Research Papers: Design Automation

Dual Residual for Centralized Augmented Lagrangian Coordination Based on Optimality Conditions

[+] Author and Article Information
Meng Xu

Mem. ASME
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: xu7@g.clemson.edu

Georges Fadel

Mem. ASME
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: fgeorge@clemson.edu

Margaret M. Wiecek

Department of Mathematical Sciences,
Clemson University,
Clemson, SC 29631
e-mail: wmalgor@clemson.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 22, 2014; final manuscript received January 29, 2015; published online March 18, 2015. Assoc. Editor: Harrison M. Kim.

J. Mech. Des 137(6), 061401 (Jun 01, 2015) (10 pages) Paper No: MD-14-1631; doi: 10.1115/1.4029788 History: Received September 22, 2014; Revised January 29, 2015; Online March 18, 2015

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.

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Figures

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Fig. 1

Procedure of centralized ALC (problems in gray areas are solved as a whole, dashed lines represent the subproblem couplings when the number of subproblem is more than 4)

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Fig. 2

The structure to solve the Golinski’s problem (a) and the micro-accelerometer problem (b) through decomposition

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Fig. 3

The optimization process of the traditional update (a) and the proposed update (b) on the speed reducer problem and the comparison of their biggest dual residuals, and (c) initial weight set by Eq. (36)

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Fig. 4

The optimization process for the traditional update (a) and the proposed update (b) on the speed reducer problem and the comparison of their biggest dual residuals, and (c) using fine-tuned initial weight

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Fig. 5

Optimization process of the centralized ALC—scheme 1 on the micro-accelerometer problem—initial weight set by Eq. (36)

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Fig. 6

Optimization process of the centralized ALC—scheme 2 on the micro-accelerometer problem—initial weight set by Eq. (36)

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Fig. 7

Optimization process of the centralized ALC—scheme 3 on the micro-accelerometer problem—initial weight set by Eq. (36)

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