In intensity modulated radiation therapy (IMRT) treatment planning beams of penetrating radiation are directed at the body from external sources. The task is to deliver enough radiation dose to designated lesion (tumor) areas without harming surrounding healthy tissue and anatomical structures. Modeling this medical objective gives rise to very large systems of inequalities. The resulting feasibility problem, which requires finding a point in the intersection of finitely many (nonempty closed and convex) sets, arises also in many other fields of science and technology.

This problem, as well as the closely related best approximation problem, can be solved by "projection algorithms" that use projections onto the individual sets and may be structurally sequential or parallel, or some combination of the two. They exhibit many interesting and useful properties.

We review the convex feasibility problem and the best approximation problem and present projection methods for their solution in light of the radiation therapy treatment planning application. In particular, we present our recent contribution to this field: the Component Averaging (CAV) algorithm.


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