Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning

Mark Brooke, Yair Censor, Aviv Gibali

Research output: Contribution to journalArticlepeer-review


We study a feasibility-seeking problem with percentage violation constraints (PVCs). These are additional constraints that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be violated by up to a specified percentage of the existing bounds. Our motivation to investigate problems with PVCs comes from the field of radiation therapy treatment planning (RTTP) wherein the fully discretized inverse planning problem is formulated as a split feasibility problem and the PVCs give rise to nonconvex constraints. Following the CQ algorithm of Byrne (2002, Inverse Problems, Vol. 18, pp. 441–53), we develop a string-averaging CQ-method that uses only projections onto the individual sets that are half-spaces represented by linear inequalities. The question of extending our theoretical results to the nonconvex sets case is still open. We describe how our results apply to RTTP and provide a numerical example.

Original languageEnglish
Number of pages25
JournalInternational Transactions in Operational Research
Issue number1
StatePublished - Jan 2023

Bibliographical note

Funding Information:
We thank Scott Penfold, Reinhard Schulte, and Frank Van den Heuvel for their help and encouragement of our work on this project. We are grateful to the reviewers for their constructive and helpful comments on the previous version of this paper. This work was supported by Cancer Research UK, grant number C2195/A25197, through a CRUK Oxford Centre DPhil Prize Studentship and by the ISF‐NSFC joint research program grant number 2874/19. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Publisher Copyright:
© 2020 The Authors. International Transactions in Operational Research published by John Wiley & Sons Ltd on behalf of International Federation of Operational Research Societies


  • CQ-algorithm
  • common fixed points
  • cutter operator
  • dose-volume constraints
  • percentage violation constraints
  • radiation therapy treatment planning
  • split feasibility
  • string-averaging

ASJC Scopus subject areas

  • Business and International Management
  • Computer Science Applications
  • Strategy and Management
  • Management Science and Operations Research
  • Management of Technology and Innovation


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