Reducing Conservatism in Pareto Robust Optimization

Abstract

Robust optimization (RO) is a sub-field of optimization theory with set-based uncertainty. A criticism of this field is that it determines optimal decisions for only the worst-case realizations of uncertainty. Several methods have been introduced to reduce this conservatism. However, non of these methods can guarantee the non-existence of another solution that improves the optimal solution for all non-worse-cases. Pareto robust optimization ensures that non-worse-case scenarios are accounted for and that the solution cannot be dominated for all scenarios. The problem with Pareto robust optimization (PRO) is that a Pareto robust optimal solution may be improved by another solution for a given subset of uncertainty. Also, Pareto robust optimal solutions are still conservative on the optimality for the worst-case scenario. In this thesis, first, we apply the concept of PRO to the Intensity Modulated Radiation Therapy (IMRT) problem. We will present a Pareto robust optimization model for four types of IMRT problems. Using several hypothetical breast cancer data sets, we show that PRO solutions decrease the side effects of overdosing while delivering the same dose that RO solutions deliver to the organs at risk. Next, we present methods to reduce the conservatism of PRO solutions. We present a method for generating alternative RO solutions for any linear robust optimization problem. We also demonstrate a method for determining if an RO solution is PRO. Then we determine the set of all PRO solutions using this method. We denote this set as the ``Pareto robust frontier" for any linear robust optimization problem. Afterward, we present a set of uncertainty realizations for which a given PRO solution is optimal. Using this approach, we compare all PRO solutions to determine the one that is optimal for the maximum number of realizations in a given set. We denote this solution as a ``superior" PRO solution for that set. At last, we introduce a method to generate a PRO solution while slightly violating the optimality of the optimal solution for the worst-case scenario. We define these solutions as ``light PRO" solutions. We illustrate the application of our approach to the IMRT problem for breast cancer. The numerical results present a significant impact of our method in reducing the side effects of radiation therapy.