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Deriving solution value bounds from the ADMM

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Abstract

This short paper describes a simple subgradient-based techniques for deriving bounds on the optimal solution value when using the ADMM to solve convex optimization problems. The technique requires a bound on the magnitude of some optimal solution vector, but is otherwise completely general. Some computational examples using LASSO problems demonstrate that the technique can produce steadily converging bounds in situations in which standard Lagrangian bounds yield little or no useful information. A second set of experiments establishes a proof of concept indicating the potential practical usefulness of the bounding technique.

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Acknowledgements

The author would like to thank Patrick Johnstone and Paulo J.S. Silva for some discussions relating to the LASSO example.

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  1. Department of Management Science and Information Systems, Rutgers Business School Newark and New Brunswick, 100 Rockafeller Road, Piscataway, NJ, 08854, USA

    Jonathan Eckstein

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  1. Jonathan Eckstein
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Correspondence to Jonathan Eckstein.

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This work was supported in part by NSF grant CCF-1617617, Computing and Communications Foundations, CISE directorate, and Air Force Office of Scientific Research (AFOSR) grant FA95550-15-1-0251.

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Eckstein, J. Deriving solution value bounds from the ADMM. Optim Lett 14, 1289–1303 (2020). https://doi.org/10.1007/s11590-020-01584-1

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  • DOI: https://doi.org/10.1007/s11590-020-01584-1

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