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

TL;DR

This paper introduces a dynamic string-averaging CQ algorithm for split feasibility problems with percentage violation constraints, motivated by radiation therapy planning, and demonstrates its applicability through numerical examples.

Contribution

It develops a novel string-averaging CQ method tailored for non-convex constraints arising in radiation therapy, extending existing algorithms to handle percentage violation constraints.

Findings

The proposed method effectively handles non-convex constraints in radiation therapy planning.

Numerical examples demonstrate the algorithm's practical applicability.

Theoretical extension to non-convex sets remains open.

Abstract

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.