A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems
We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, fo...
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| Veröffentlicht in: | Mathematical programming Jg. 175; H. 1-2; S. 503 - 536 |
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| Abstract | We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. We adapt the augmented Lagrangian method framework to address the presence of nonconvexity in the non-relaxed constraint set and to enable efficient parallelization. The development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions. However, deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized. Pivotal in our modification to the augmented Lagrangian method is an integration of the simplicial decomposition method and the nonlinear block Gauss–Seidel method. An adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted. Under mild conditions optimal dual convergence is proven, and we report computational results on test instances from the stochastic optimization literature. We demonstrate improvement in parallel speedup over a baseline parallel approach. |
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| AbstractList | We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. We adapt the augmented Lagrangian method framework to address the presence of nonconvexity in the non-relaxed constraint set and to enable efficient parallelization. The development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions. However, deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized. Pivotal in our modification to the augmented Lagrangian method is an integration of the simplicial decomposition method and the nonlinear block Gauss–Seidel method. An adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted. Under mild conditions optimal dual convergence is proven, and we report computational results on test instances from the stochastic optimization literature. We demonstrate improvement in parallel speedup over a baseline parallel approach. |
| Author | Dandurand, Brian Eberhard, Andrew Oliveira, Fabricio Boland, Natashia Christiansen, Jeffrey |
| Author_xml | – sequence: 1 givenname: Natashia surname: Boland fullname: Boland, Natashia organization: Georgia Institute of Technology – sequence: 2 givenname: Jeffrey surname: Christiansen fullname: Christiansen, Jeffrey organization: RMIT University – sequence: 3 givenname: Brian surname: Dandurand fullname: Dandurand, Brian organization: RMIT University – sequence: 4 givenname: Andrew surname: Eberhard fullname: Eberhard, Andrew email: andy.eberhard@rmit.edu.au organization: RMIT University – sequence: 5 givenname: Fabricio surname: Oliveira fullname: Oliveira, Fabricio organization: Aalto University |
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| DOI | 10.1007/s10107-018-1253-9 |
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| Keywords | Nonlinear block Gauss–Seidel method Augmented Lagrangian method Proximal bundle method 90-08 Simplicial decomposition method 90C06 90C15 90C26 Parallel computing 90C30 90C46 90C25 90C11 |
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