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

Mark Brooke(mark.brooke***at***wolfson.ox.ac.uk)
Yair Censor(yair***at***math.haifa.ac.il)
Aviv Gibali(avivg***at***braude.ac.il)

Abstract: We study a feasibility-seeking problem with percentage violation constraints. 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 percentage violation constraints comes from the field of radiation therapy treatment planning wherein the fully-discretized inverse planning problem is formulated as a split feasibility problem and the percentage violation constraints give rise to non-convex constraints. We develop a string-averaging CQ method that uses only projections onto the individual sets which are half-spaces represented by linear inequalities. The question of extending our theoretical results to the non-convex sets case is still open. We describe how our results apply to radiation therapy treatment planning and provide a numerical example.

Keywords: String-averaging, CQ-algorithm, split feasibility, percentage violation constraints, radiation therapy treatment planning, dose-volume constraints, common fixed points, cutter operator.

Category 1: Applications -- Science and Engineering (Biomedical Applications )

Category 2: Nonlinear Optimization

Citation: International Transactions in Operational Research (2020). Published online: December 30, 2020. Available at https://onlinelibrary.wiley.com/doi/full/10.1111/itor.12929.

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Entry Submitted: 01/04/2021
Entry Accepted: 01/04/2021
Entry Last Modified: 01/04/2021

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