Perturbed Projection Methods in Convex Optimization - Applied to Radiotherapy Planning

Projection methods frequently suffer from zigzagging behavior and therefore slow convergence when applied to ill-conditioned optimization problems, for example those arising in radiotherapy treatment planning. In this work we propose to exploit the bounded perturbation resilience of the projection methods and use perturbations as a tool to circumvent the zigzagging behavior, thereby accelerating the convergence significantly.

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This thesis is motivated by the treatment planning problem in intensity modulated radiation therapy (IMRT). We tackle the multicriteria optimization problem arising from this application by transforming it into a sequence of convex feasibility problems via the level set scheme and then solve each feasibility problem using projection methods. Some characteristics of the IMRT treatment planning problem are challenging to this strategy. Ill-conditionedness and the correlation of the objective functions often lead to zigzagging behavior by the projection methods and therefore slow convergence of the overall optimization procedure. To mitigate these disadvantages, we exploit the bounded perturbation resilience of the projection methods. We introduce three new perturbations designed to avoid the zigzagging behavior and combine them with the projection methods. We study both the theoretical and computational impact of the suggested perturbed iteration schemes. We demonstrate our methods on linear examples and also apply them to nonlinear optimization problems arising from IMRT treatment planning on real cases.