Interior Point Methods Can Exploit Structure of Convex Piecewise Linear Functions with Application in Radiation Therapy

Abstract

Auxiliary variables are often used to model a convex piecewise linear function in the framework of linear optimization. This work shows that such variables yield a block diagonal plus low rank structure in the reduced KKT system of the dual problem. We show how the structure can be detected efficiently and derive the linear algebra formulas for an interior point method which exploits such a structure. The structure is detected in 36% of the cases in Netlib. Numerical results on the inverse planning problem in radiation therapy show an order of magnitude speed-up compared to the state-of-the-art interior point solver CPLEX and considerable improvements in dose distribution compared to current algorithms.