On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

Yair Censor, Wei Chen, Patrick L. Combettes, Ran Davidi, Gabor T. Herman, W. Chen, R. Davidi, G. T. Herman, P. L. Combettes

Research output: Contribution to journalArticlepeer-review


The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many real-world applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints).

Original languageEnglish
Pages (from-to)1065-1088
Number of pages24
JournalComputational Optimization and Applications
Issue number3
StatePublished - Apr 2012

Bibliographical note

Funding Information:
Acknowledgements The work of Y. Censor, W. Chen, R. Davidi, and G.T. Herman was supported by NIH Award Number R01HL070472 from the National Heart, Lung, and Blood Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Heart, Lung, and Blood Institute or the National Institutes of Health. The work of P.L. Combettes was supported by the Agence Nationale de la Recherche under grant ANR-08-BLAN-0294-02. The work of Y. Censor was also supported by United States-Israel Binational Science Foundation (BSF) Grant number 200912. We thank the Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical School (and especially David Craft) for providing access to their computers. We are also grateful to T. Goldstein and S. Osher of UCLA for providing us with the optimization code [51] that they considered most appropriate for solving (24).


  • Convex feasibility problems
  • Linear inequalities
  • Numerical evaluation
  • Optimization
  • Projection methods
  • Sparse matrices

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


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