Multi-Criteria Inverse Robustness in Radiotherapy Planning Using Semidefinite Programming

Radiotherapy planning naturally leads to a multi-criteria optimization problem which is subject to different sources of uncertainty. In order to find the desired treatment plan, a decision maker must balance these objectives as well as the level of robustness towards uncertainty against each other. This paper showcases a quantitative approach to do so, which combines the theoretical model with the ability to deal with practical challenges. To this end, the uncertainty, which can be expressed via the so-called dose-influence matrix, is modelled using interval matrices. We use inverse robustness to introduce an additional objective, which aims to maximize the volume of the uncertainty set. A multi-criteria approach allows to handle the uncertainty while keeping appropriate values of the other objective functions. We solve the resulting quadratically constrained quadratic optimization problem (QCQP) by first relaxing it to a convex semidefinite problem (SDP) and then reconstructing optimal solutions of the QCQP from solutions of the SDP.