scholarly journals An augmented Lagrangian coordination-decomposition algorithm for solving distributed non-convex programs

2014 ◽  
Author(s):  
Jean-Hubert Hours ◽  
Colin N. Jones
Keyword(s):  
Augmented Lagrangian ◽  
Decomposition Algorithm ◽  
Convex Programs ◽  
Augmented Lagrangian Coordination

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.

scholarly journals Improved augmented Lagrangian coordination for optimizing supply chain configuration with multiple sharing elements in industrial cluster

2019 ◽  
Vol 119 (4) ◽  
pp. 743-773 ◽  
Author(s):  
Duxian Nie ◽  
Ting Qu ◽  
Yang Liu ◽  
Congdong Li ◽  
G.Q. Huang
Keyword(s):  
Supply Chain ◽  
Augmented Lagrangian ◽  
Industrial Cluster ◽  
Sales Price ◽  
Content Type ◽  
Cooperative Strategies ◽  
Capacity Sharing ◽  
Supply Chain Configuration ◽  
Augmented Lagrangian Coordination ◽  
Chain Configuration

Purpose The purpose of this paper is to study various combination forms of the three basic sharing elements (i.e. orders sharing, manufacturers capacity sharing and suppliers capacity sharing) in the cluster supply chain (CSC), formulate a distributed model to protect enterprises’ decision privacy and seek to develop an effective method for solving the distributed complex model. Design/methodology/approach A distributed assembly cluster supply chain configuration (ACSCC) model is formulated. An improved augmented Lagrangian coordination (ALC) is proposed and used to solve the ACSCC model. A series of experiments are conducted to validate the improved ALC and the model. Findings Two major findings are obtained. First, the market order’s quantity change and the sales price of the product have a great impact on both the optimal results of the ACSCC and the cooperative strategy, especially, when the market order increases sharply, enterprises have to adopt multiple cooperative strategies to complete the order; meanwhile, the lower sales price of the product helps independent suppliers to get more orders. Second, the efficiency and computational accuracy of the improved ALC method are validated as compared to the centralized ALC and Lingo11. Research limitations/implications This paper formulated the single-period ACSCC model under certain assumptions, yet a multi-period ACSCC model is to be developed, a more comprehensive investigation of the relationships among combination forms is to be extended further and a rigid proof of the improved ALC is necessary. Practical implications Enterprises in the industrial cluster should adopt different cooperative strategies in terms of the market order’s quantity change and the sales price of the product. Social implications The proposed various combination forms of sharing elements and the formulated ACSCC model provide guidance to managers in the industrial cluster to choose the proper policy. Originality/value This research studies various combination forms of the three basic sharing elements in the CSC. A distributed ACSCC model has been established considering simultaneously multiple sharing elements. An improved ALC is presented and applied to the ACSCC 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 Distributed Augmented Lagrangian Coordination Based on Optimality Conditions

2015 ◽  
Author(s):  
Meng Xu ◽  
Georges Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Optimality Conditions ◽  
Augmented Lagrangian ◽  
Engineering Problem ◽  
Process Efficiency ◽  
Master Problem ◽  
Coordination Strategies ◽  
Augmented Lagrangian Relaxation ◽  
Mathematical Problems ◽  
Update Strategy ◽  
Augmented Lagrangian Coordination

Augmented Lagrangian Coordination (ALC) is one of the more popular coordination strategies for decomposition based optimization. It employs the augmented Lagrangian relaxation approach and has shown great improvements in terms of efficiency and solution accuracy when compared to other methods addressing the same type of problem. Additionally, by offering two variants: the centralized ALC in which an artificial master problem in the upper level is created to coordinate all the sub-problems in the lower level, and the distributed ALC in which coordination can be performed directly between sub-problems without a master problem, ALC provides more flexibility than other methods. However, the initial setting and the update strategy of the penalty weights in ALC still significantly affect its performance and thus are worth further research. For centralized ALC, the non-monotone weight update strategy based on the theory of dual residual has shown very good improvements over the traditional monotone update, in which the penalty weights can either increase or decrease. In this paper, we extend the research on the dual residual in centralized ALC to the distributed ALC. Through applying the Karush-Kuhn-Tucker (KKT) optimality conditions to the All-In-One (AIO) and decomposed problems, the necessary conditions for the decomposed solution to be optimal are derived, which leads to the definition of primal and dual residuals in distributed ALC. A new non-monotone weight based on both residuals is then proposed, by which all AIO KKT conditions are guaranteed after decomposition. Numerical tests are conducted on two mathematical problems and one engineering problem and the performances of the new update are compared to those of the traditional update. The results show that our proposed methods improve the process efficiency, accuracy, and robustness for distributed ALC.

Dual Residual in Augmented Lagrangian Coordination for Decomposition-Based Optimization

2014 ◽  
Author(s):  
Meng Xu ◽  
Georges Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Augmented Lagrangian ◽  
Test Problems ◽  
Initial Weight ◽  
Penalty Term ◽  
Augmented Lagrangian Relaxation ◽  
Primal And Dual ◽  
Update Strategy ◽  
Solution Accuracy ◽  
Penalty Weight ◽  
Augmented Lagrangian Coordination

As system design problems increase in complexity, researchers seek approaches to optimize such problems by coordinating the optimizations of decomposed sub-problems. Many methods for optimization by decomposition have been proposed in the literature among which, the Augmented Lagrangian Coordination (ALC) method has drawn much attention due to its efficiency and flexibility. The ALC method involves a quadratic penalty term, and the initial setting and update strategy of the penalty weight are critical to the performance of the ALC. The weight in the traditional weight update strategy always increases and previous research shows that an inappropriate initial value of the penalty weight may cause the method not to converge to optimal solutions. Inspired by the research on Augmented Lagrangian Relaxation in the convex optimization area, a new weight update strategy in which the weight can either increase or decrease is introduced into engineering optimization. The derivation of the primal and dual residuals for optimization by decomposition is conducted as a first step. It shows that the traditional weight update strategy only considers the primal residual, which may result in a duality gap and cause a relatively big solution error. A new weight update strategy considering both the primal and dual residuals is developed which drives the dual residual to zero in the optimization process, thus guaranteeing the solution accuracy of the decomposed problem. Finally, the developed strategy is applied to both mathematical and engineering test problems and the results show significant improvements in solution accuracy. Additionally, the proposed approach makes the ALC method more robust since it allows the coordination to converge with an initial weight selected from a much wider range of possible values while the selection of initial weight is a big concern in the traditional weight update strategy.

Consistency Constraint Allocation in Augmented Lagrangian Coordination

2010 ◽  
Vol 132 (7) ◽  
Author(s):  
James T. Allison ◽  
Panos Y. Papalambros
Keyword(s):  
System Design ◽  
Design Problem ◽  
Augmented Lagrangian ◽  
Structure Theory ◽  
Additional Element ◽  
Pump Design ◽  
Coordination Strategy ◽  
Ideal System ◽  
Consistency Constraint ◽  
Augmented Lagrangian Coordination

Many engineering systems are too complex to design as a single entity. Decomposition-based design optimization methods partition a system design problem into subproblems, and coordinate subproblem solutions toward an optimal system design. Recent work has addressed formal methods for determining an ideal system partition and coordination strategy, but coordination decisions have been limited to subproblem sequencing. An additional element in a coordination strategy is the linking structure of the partitioned problem, i.e., the allocation of constraints that guarantee that the linking variables among subproblems are consistent. There may exist many alternative linking structures for a decomposition-based strategy that can be selected for a given partition, and this selection should be part of an optimal simultaneous partitioning and coordination scheme. This article develops a linking structure theory for a particular class of decomposition-based optimization algorithms, augmented Lagrangian coordination (ALC). A new formulation and coordination technique for parallel ALC implementations is introduced along with a specific linking structure theory, yielding a partitioning and coordination selection method for ALC that includes consistency constraint allocation. This method is demonstrated using an electric water pump design problem.

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.

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.

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