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Using inexact gradients in a multilevel optimization algorithm

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Abstract

Many optimization algorithms require gradients of the model functions, but computing accurate gradients can be computationally expensive. We study the implications of using inexact gradients in the context of the multilevel optimization algorithm MG/Opt. MG/Opt recursively uses (typically cheaper) coarse models to obtain search directions for finer-level models. However, MG/Opt requires the gradient on the fine level to define the recursion. Our primary focus here is the impact of the gradient errors on the multilevel recursion. We analyze, partly through model problems, how MG/Opt is affected under various assumptions about the source of the error in the gradients, and demonstrate that in many cases the effect of the errors is benign. Computational experiments are included.

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Acknowledgements

The research was supported by the US Department of Energy under Award DE-SC-0001691. We thank Paul Boggs and David Gay for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, College of William & Mary, Williamsburg, VA, 23185, USA

    Robert Michael Lewis

  2. Systems Engineering and Operations Research Department, George Mason University, Fairfax, VA, 22030, USA

    Stephen G. Nash

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  1. Robert Michael Lewis
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  2. Stephen G. Nash
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Correspondence to Stephen G. Nash.

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Lewis, R.M., Nash, S.G. Using inexact gradients in a multilevel optimization algorithm. Comput Optim Appl 56, 39–61 (2013). https://doi.org/10.1007/s10589-013-9546-7

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