| The primary purpose of this study is to develop efficient optimization techniques for solving radiation treatment planning problems. We consider a number of optimization formulations, both linear and nonlinear, and study computational issues that arise in this field. The goal is to identify and develop the most efficient method for solving each formulation. The results include an extensive computational study of linear and quadratic problems that arise in treatment planning, a new methodology to solve the nonlinear problem arising from using equivalent uniform dose (EUD) in the objective, and finally a novel approach for formulating and solving problems in which the uncertainty in the data is explicitly accounted for.;Many approaches use linear programming formulations as a core computation. However, these formulations of treatment planning often require a surprisingly large amount of time to solve and the choices of formulation, algorithm, and pivot rule that perform best from a computational viewpoint are not always obvious. We consider several linear programming formulations of treatment planning problem and test several variants of simplex and interior-point methods for solving them.;Secondly, we consider treatment planning via biological objectives. Particularly, we consider a formulation with nonlinear objectives based on EUD, with bound constraints, and describe fast, flexible variants of the two-metric gradient projection (2MGP) approach for solving them efficiently. We also present a novel heuristic for obtaining an approximate solution with a smoother distribution of beamlet weights. The variant of the 2MGP approach that computes the Newton part of each step iteratively using modified conjugate gradients, using an implicit representation of the Hessian, is most effective. Also, the beamlet weights can be smoothed efficiently, with only minor degradation of the solution's quality.;Finally, we use robust optimization techniques to formulate a treatment planning problem where both errors from the dose calculation and inter-fraction positional uncertainty of tumor and organs are taken into account. We describe an efficient approach for solving the resulting second-order cone program. We show that robust optimization can be implemented effectively at a computational cost not much greater than conventional approaches that account for uncertainty in cruder ways. |
