Incorporating Radial Basis Functions in Pattern Search Methods: Application to Beam Angle Optimization in Radiotherapy Treatment Planning

The global optimization of black-box functions with many local minima occurs in many branches of science and engineering. There are many methods and heuristics to address this type of problems. However, for problems with expensive black-box functions, both in terms of cost or time, the number of function evaluations required by most of the methods or heuristics is prohibitive. The pattern search methods framework is suited to address this type of problems since it requires few function value evaluations to converge and have the ability to avoid local entrapment. The ability of this class of methods to obtain global minima depends on the incorporation of methods or heuristics for global optimization on their, so called, search step. We propose the use of radial basis functions both to influence the quality of the local minimizer found by the method and also to obtain a better coverage of the search space. Our approach is tailored for addressing the beam angle optimization (BAO) problem in intensity modulated radiation therapy treatment planning, but can be easily extended for other general problems. The BAO problem is quite difficult, and yet to be solved in a satisfactory way, since it is a highly non-convex optimization problem with many local minima. A couple of retrospective treated cases of head-and-neck tumors at the Portuguese Institute of Oncology of Coimbra is used to discuss the benefits of using our approach in the optimization of the BAO problem.