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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Bibliographic Details
Published in:	Mathematical programming Vol. 175; no. 1-2; pp. 503 - 536
Main Authors:	Boland, Natashia, Christiansen, Jeffrey, Dandurand, Brian, Eberhard, Andrew, Oliveira, Fabricio
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2019 Springer Nature B.V
Subjects:	Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Gauss-Seidel method Lagrange multiplier Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Parallel processing Theoretical 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
ISSN:	0025-5610, 1436-4646
Online Access:	Get full text
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Summary:	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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ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-018-1253-9