In this paper we study a feasibility-seeking problem with percentage violation con- straints. 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 specied fraction of the number of constraints in each subset is allowed to be violated by up to a specied percentage of the existing bounds. Our motiva- tion to investigate problems with percentage violation constraints comes from the eld of radiation therapy treatment planning wherein the fully-discretized inverse planning problem is formulated as a split feasibility problem and the percentage violation con- straints 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.
Preprint, November 2019.