Minimal value of the maximal dose fraction in the optimization of the radiotherapy scheduling

We report the analytical study of a constrained optimization problem which consists in finding the minimal value of the largest entry of a vector d in R^5, with constraints involving the entries sum and the squared entries sum. The solution of the problem studied in this paper provides the proof of a property needed to determine the admissible protocols of the radiotherapy scheduling optimization problem presented in Bruni et al (2014), that includes constraints limiting the radiation damages to normal tissues. In particular, we nd the minimal value of the maximal dose fraction of the protocols producing the maximal tolerable damage to both early and late responding tissues when there is not a prevalent normal tissue constraint. Then, we extend the property to all the protocols producing the maximal damage to the late responding tissue only.