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.

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.

scholarly journals Benders decomposition with adaptive oracles for large scale optimization

2020 ◽  
Author(s):  
Nicolò Mazzi ◽  
Andreas Grothey ◽  
Ken McKinnon ◽  
Nagisa Sugishita
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Computational Effort ◽  
Planning Problem ◽  
Illustrative Case ◽  
Investment Planning ◽  
Cutting Plane Algorithms ◽  
Inexact Information

AbstractThis paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an $$\epsilon $$ ϵ -optimal solution: an illustrative case is 31.9 times faster for a $$1.00\%$$ 1.00 % convergence tolerance and 15.4 times faster for a $$0.01\%$$ 0.01 % tolerance.

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.

scholarly journals TRACING THE EMERGENCE OF DESIGN PROBLEMS AND THEIR IMPACTS ON THE COMPLEXITY OF ENGINEERING SOLUTIONS

2021 ◽  
Vol 1 ◽  
pp. 3229-3238
Author(s):  
Torben Beernaert ◽  
Pascal Etman ◽  
Maarten De Bock ◽  
Ivo Classen ◽  
Marco De Baar
Keyword(s):  
Large Scale ◽  
Fusion Reactor ◽  
Design Stage ◽  
Engineering Model ◽  
Design Problems ◽  
Nuclear Fusion Reactor ◽  
Early Design Stage ◽  
Iterative Design ◽  
Problems And Solutions ◽  
Emergent Design

AbstractThe design of ITER, a large-scale nuclear fusion reactor, is intertwined with profound research and development efforts. Tough problems call for novel solutions, but the low maturity of those solutions can lead to unexpected problems. If designers keep solving such emergent problems in iterative design cycles, the complexity of the resulting design is bound to increase. Instead, we want to show designers the sources of emergent design problems, so they may be dealt with more effectively. We propose to model the interplay between multiple problems and solutions in a problem network. Each problem and solution is then connected to a dynamically changing engineering model, a graph of physical components. By analysing the problem network and the engineering model, we can (1) derive which problem has emerged from which solution and (2) compute the contribution of each design effort to the complexity of the evolving engineering model. The method is demonstrated for a sequence of problems and solutions that characterized the early design stage of an optical subsystem of ITER.

scholarly journals On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(β-alanine)-Electrolyte

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 113
Author(s):  
Stephan Daniel Schwoebel ◽  
Thomas Mehner ◽  
Thomas Lampke
Keyword(s):  
Augmented Lagrangian ◽  
Computation Time ◽  
Species Distributions ◽  
Chemical Processes ◽  
Diffusion Reaction ◽  
Augmented Lagrangian Methods ◽  
Major Question ◽  
Component Systems ◽  
Reaction Models ◽  
Reaction Equations

Three-component systems of diffusion–reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion–reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(β-alanine) electrolyte are presented.

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