Skip to main content

Advertisement

Springer Nature Link
Log in

A computation study on an integrated alternating direction method of multipliers for large scale optimization

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

The alternating direction method of multipliers (ADMM) has recently received a lot of attention especially due to its capability to harness the power of the new parallel and distributed computing environments. However, ADMM could be notoriously slow especially if the penalty parameter, assigned to the augmented term in the objective function, is not properly chosen. This paper aims to accelerate ADMM by integrating that with the Barzilai–Borwein gradient method and an acceleration technique known as line search. Line search accelerates an iterative method by performing a one-dimensional search along the line segment connecting two successive iterations. We pay a special attention to the large-scale nonnegative least squares problems, and our experiments using real datasets indicate that the integration not only accelerate ADMM but also robustifies that against the penalty parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. For computational efficiency, the code does not include a diminishing stepsize that was suggested in the paper to assure the convergence.

References

  1. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H., Deutsch, F., Hundal, H., Park, S.H.: Accelerating the convergence of the method of alternating projections. Trans. Am. Math. Soc. 355(9), 3433–3461 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG—software for convex-constrained optimization. ACM Trans. Math. Softw. (TOMS) 27(3), 340–349 (2001)

    Article  MATH  Google Scholar 

  4. Björck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

  5. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)

    Article  MATH  Google Scholar 

  6. Bro, R., De Jong, S.: A fast non-negativity-constrained least squares algorithm. J. Chemom. 11(5), 393–401 (1997)

    Article  Google Scholar 

  7. Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190–1208 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, D., Plemmons, R.J.: Nonnegativity constraints in numerical analysis. In: Symposium on the Birth of Numerical Analysis, pp. 109–140 (2009)

  9. Craft, D., McQuaid, D., Wala, J., Chen, W., Salari, E., Bortfeld, T.: Multicriteria VMAT optimization. Med. Phys. 39(2), 686–696 (2012)

    Article  Google Scholar 

  10. Dai, Y.H., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100(1), 21–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dai, Y.H., Liao, L.Z.: R-linear convergence of the barzilai and borwein gradient method. IMA J. Numer. Anal. 22(1), 1–10 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dax, A.: Successive refinement of large multicell models. SIAM J. Numer. Anal. 22(5), 865–887 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dax, A.: Line search acceleration of iterative methods. Linear Algebra Appl. 130, 43–63 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. De Pierro, A., Lopes, J.: Accelerating iterative algorithms for symmetric linear complementarity problems. Int. J. Comput. Math. 50(1–2), 35–44 (1994)

    Article  MATH  Google Scholar 

  15. Deasy, J.O., Blanco, A.I., Clark, V.H.: CERR: a computational environment for radiotherapy research. Med. Phys. 30(5), 979–985 (2003)

    Article  Google Scholar 

  16. Deutsch, F.: Accelerating the convergence of the method of alternating projections via a line search: a brief survey. Stud. Comput. Math. 8, 203–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eckstein, J.: Augmented Lagrangian and alternating direction methods for convex optimization: a tutorial and some illustrative computational results. RUTCOR Research Report RRR 32–2012, (2012)

  18. Ehrgott, M., Güler, Ç., Hamacher, H.W., Shao, L.: Mathematical optimization in intensity modulated radiation therapy. 4OR 6(3), 199–262 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fang, E.X., He, B., Liu, H., Yuan, X.: Generalized alternating direction method of multipliers: new theoretical insights and applications. Math. Program. Comput. 7(2), 149–187 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fletcher, R.: Low storage methods for unconstrained optimization. Lect. Appl. Math. (AMS) 26, 165–179 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Fletcher, R.: On the Barzilai–Borwein method. In: Qi, L., Teo, K., Yang, X. (eds.) Optimization and Control with Applications, pp. 235–256. Springer, New York (2005)

  22. Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Elsevier, Amsterdam (2000)

    MATH  Google Scholar 

  23. Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 1(1), 93–111 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Glunt, W., Hayden, T., Raydan, M.: Molecular conformations from distance matrices. J. Comput. Chem. 14(1), 114–120 (1993)

    Article  Google Scholar 

  25. Glunt, W., Hayden, T.L., Raydan, M.: Preconditioners for distance matrix algorithms. J. Comput. Chem. 15(2), 227–232 (1994)

    Article  Google Scholar 

  26. He, B., Yang, H.: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23(3), 151–161 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. He, B., Yang, H., Wang, S.: Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theory Appl. 106(2), 337–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kim, D., Sra, S., Dhillon, I.S.: A non-monotonic method for large-scale non-negative least squares. Optim. Methods Softw. 28(5), 1012–1039 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kontogiorgis, S., Meyer, R.R.: A variable-penalty alternating directions method for convex optimization. Math. Program. 83(1–3), 29–53 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM, Philadelphia, (1995)

  31. Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, vol. 116. Springer, Berlin (2008)

    MATH  Google Scholar 

  32. Molina, B., Raydan, M.: Preconditioned Barzilai–Borwein method for the numerical solution of partial differential equations. Numer. Algorithms 13(1), 45–60 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Raydan, M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13(3), 321–326 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7(1), 26–33 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shepard, D.M., Ferris, M.C., Olivera, G.H., Mackie, T.R.: Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev. 41(4), 721–744 (1999)

    Article  MATH  Google Scholar 

  36. Xing, L., Hamilton, R., Spelbring, D., Pelizzari, C., Chen, G., Boyer, A.: Fast iterative algorithms for three-dimensional inverse treatment planning. Med. Phys. 25(10), 1845–1849 (1998)

    Article  Google Scholar 

  37. Xing, L., Li, R.: Inverse planning in the age of digital linacs: station parameter optimized radiation therapy (sport). J. Phys. Conf. Ser. 489, 12065–12070 (2014)

    Article  Google Scholar 

  38. Yang, Y., Xing, L.: Clinical knowledge-based inverse treatment planning. Phys. Med. Biol. 49(22), 5101 (2004)

    Article  Google Scholar 

  39. Yu, C.X.: Intensity-modulated arc therapy with dynamic multileaf collimation: an alternative to tomotherapy. Phys. Med. Biol. 40(9), 1435 (1995)

    Article  Google Scholar 

  40. Zarepisheh, M., Li, R., Ye, Y., Xing, L.: Simultaneous beam sampling and aperture shape optimization for sport. Med. Phys. 42(2), 1012–1022 (2015)

    Article  Google Scholar 

  41. Zarepisheh, M., Long, T., Li, N., Tian, Z., Romeijn, H.E., Jia, X., Jiang, S.B.: A DVH-guided IMRT optimization algorithm for automatic treatment planning and adaptive radiotherapy replanning. Med. Phys. 41(6), 061,711 (2014)

    Article  Google Scholar 

  42. Zarepisheh, M., Uribe-Sanchez, A.F., Li, N., Jia, X., Jiang, S.B.: A multicriteria framework with voxel-dependent parameters for radiotherapy treatment plan optimization. Med. Phys. 41(4), 041,705 (2014)

    Article  Google Scholar 

  43. Zhang, J., Morini, B.: Solving regularized linear least-squares problems by the alternating direction method with applications to image restoration. Electron. Trans. Numer. Anal. 40, 356–372 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by NIH (1R01 CA176553 and R01E0116777).

Author information

Authors and Affiliations

  1. Department of Medical Physics, Memorial Sloan Kettering Cancer Center, New York, NY, USA

    Masoud Zarepisheh

  2. Department of Radiation Oncology, Stanford University, Stanford, CA, USA

    Lei Xing

  3. Department of Management Science and Engineering, Stanford University, Stanford, CA, USA

    Yinyu Ye

Authors
  1. Masoud Zarepisheh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yinyu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masoud Zarepisheh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zarepisheh, M., Xing, L. & Ye, Y. A computation study on an integrated alternating direction method of multipliers for large scale optimization. Optim Lett 12, 3–15 (2018). https://doi.org/10.1007/s11590-017-1116-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1116-y

Keywords

Search

Navigation