A two-variable linear program solves the standard linear–quadratic formulation of the fractionation problem in cancer radiotherapy

The standard formulation of the fractionation problem with multiple organs-at-risk based on the linear–quadratic dose–response model requires the solution of a nonconvex quadratically constrained quadratic program. Existing literature therefore uses heuristic methods without any analyses about solution quality. There is no known method that is guaranteed to find an optimal solution. We prove that this formulation of the fractionation problem can in fact be solved to optimality by instead solving a two-variable linear program with a few constraints.