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Accelerating Two Projection Methods via Perturbations with Application to Intensity-Modulated Radiation Therapy

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Abstract

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of k-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower.

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Acknowledgements

We wish to thank the editor and the anonymous referees for the thorough analysis and review, their comments and suggestions helped tremendously in improving the quality of this paper and made it suitable for publication.

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Authors and Affiliations

  1. Optimization Department, Fraunhofer ITWM, 67663, Kaiserslautern, Germany

    Esther Bonacker & Karl-Heinz Küfer

  2. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  3. The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838, Haifa, Israel

    Aviv Gibali

Authors
  1. Esther Bonacker
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  2. Aviv Gibali
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  3. Karl-Heinz Küfer
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Correspondence to Aviv Gibali.

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Bonacker, E., Gibali, A. & Küfer, KH. Accelerating Two Projection Methods via Perturbations with Application to Intensity-Modulated Radiation Therapy. Appl Math Optim 83, 881–914 (2021). https://doi.org/10.1007/s00245-019-09571-4

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