An Interior Point Constraint Generation Algorithm for Semi-Infinite Optimization with Health-Care Application

Abstract

We propose an interior point constraint generation (IPCG) algorithm for semi-infinite linear optimization (SILO) and prove that the algorithm converges to an ε-solution of SILO after a finite number of constraints is generated. We derive a complexity bound on the number of Newton steps needed to approach the updated μ-center after adding multiple violated constraints and a complexity bound on the total number of constraints that is required for the overall algorithm to converge. We implement our algorithm to solve the sector duration optimization problem arising in Leksell Gamma Knife® Perfexion™ (Elekta, Stockholm Sweden) treatment planning, a highly specialized treatment for brain tumors. Using real patient data provided by the Department of Radiation Oncology at Princess Margaret Hospital in Toronto, Ontario, Canada, we show that our algorithm can efficiently handle problems in real-life health-care applications by providing a quality treatment plan in a timely manner. Comparing our computational results with MOSEK, a commercial software package, we show that the IPCG algorithm outperforms the classical primal-dual interior point methods on sector duration optimization problem arising in Perfexion™ treatment planning. We also compare our results with that of a projected gradient method. In both cases we show that IPCG algorithm obtains a more accurate solution substantially faster.