Parallel subgradient algorithm with block dual decomposition for large-scale optimization
•Trade-off between minimizing the number of dualized constraints and detecting a structure amenable to parallel optimization.•Our approach accelerates the convergence of the distributed sub-gradient method when compared to the dual decomposition.•Denser constraint matrix leads to a higher number of...
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| Vydáno v: | European journal of operational research Ročník 299; číslo 1; s. 60 - 74 |
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| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Netherlands
Elsevier B.V
16.05.2022
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| ISSN: | 0377-2217, 1872-6860 |
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| Abstract | •Trade-off between minimizing the number of dualized constraints and detecting a structure amenable to parallel optimization.•Our approach accelerates the convergence of the distributed sub-gradient method when compared to the dual decomposition.•Denser constraint matrix leads to a higher number of dualized constraints and more iterations for convergence.•It is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems.
This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition. |
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| AbstractList | This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition.This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition. This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition. •Trade-off between minimizing the number of dualized constraints and detecting a structure amenable to parallel optimization.•Our approach accelerates the convergence of the distributed sub-gradient method when compared to the dual decomposition.•Denser constraint matrix leads to a higher number of dualized constraints and more iterations for convergence.•It is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems. This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition. |
| Author | Zheng, Yuchen Serban, Nicoleta Lee, Ilbin Dehghanian, Amin Xie, Yujia |
| AuthorAffiliation | b Alberta School of Business, University of Alberta, 2-29B Business Building, Edmonton, Alberta T6G 2R6, Canada a H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW Atlanta, GA 30332 |
| AuthorAffiliation_xml | – name: b Alberta School of Business, University of Alberta, 2-29B Business Building, Edmonton, Alberta T6G 2R6, Canada – name: a H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW Atlanta, GA 30332 |
| Author_xml | – sequence: 1 givenname: Yuchen surname: Zheng fullname: Zheng, Yuchen email: yzheng67@gmail.com organization: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW, Atlanta, GA 30332, USA – sequence: 2 givenname: Yujia surname: Xie fullname: Xie, Yujia email: yxie@gatech.edu organization: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW, Atlanta, GA 30332, USA – sequence: 3 givenname: Ilbin orcidid: 0000-0003-1323-0393 surname: Lee fullname: Lee, Ilbin email: ilbin@ualberta.ca organization: Alberta School of Business, University of Alberta, 2-29B Business Building, Edmonton, Alberta T6G 2R6, Canada – sequence: 4 givenname: Amin surname: Dehghanian fullname: Dehghanian, Amin email: adehghanian3@gatech.edu organization: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW, Atlanta, GA 30332, USA – sequence: 5 givenname: Nicoleta orcidid: 0000-0002-5813-7435 surname: Serban fullname: Serban, Nicoleta email: nserban@isye.gatech.edu organization: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Dr. NW, Atlanta, GA 30332, USA |
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| Keywords | Block dual decomposition Large scale optimization Community detection Parallel subgradient algorithm Distributed decision making Block Dual Decomposition Parallel Subgradient Algorithm |
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| SubjectTerms | Block dual decomposition Community detection Distributed decision making Large scale optimization Parallel subgradient algorithm |
| Title | Parallel subgradient algorithm with block dual decomposition for large-scale optimization |
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