  In this paper 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. 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
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.
