Multi-objective optimisation of positively homogeneous functions and an application in radiation therapy

Unconstrained multi-objective optimisation problems with p positively homogeneous objective functions are considered. We prove that such problems reduce to multi-objective optimisation problems with p−1 objectives and a single equality constraint. Thus, problems with two objectives can be solved with standard single objective optimisation methods and, for problems with p>2 objectives, we can compute infinitely many efficient solutions by solving a finite number of single objective problems. The proposed procedure is applied on radiotherapy for cancer treatment.