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

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.