Preconditioned CG and GMRES for interior point methods with applications in radiation therapy

Abstract

Interior point methods (IPMs) are widely used for different types of mathematical optimization problems. Many implementations of IPMs in use today rely on direct linear solvers to solve systems of equations in each iteration. The need to solve ever larger optimization problems more efficiently and the rise of hardware accelerators for general-purpose computing has led to much interest in using iterative linear solvers instead, though inevitable ill-conditioning of the linear systems remains a challenge. We investigate the use of the Krylov solvers CG, MINRES and GMRES for IPMs applied to optimization problems from radiation therapy. We implement a prototype IPM for convex quadratic programs and consider two different preconditioning strategies depending on the characteristics of the problem. One where a doubly augmented re-formulation of the Karush–Kuhn–Tucker linear system, originally proposed by Forsgren and Gill, is used together with a Jacobi-preconditioned conjugate gradient solver, and another with constraint preconditioning applied to a symmetric indefinite formulation of the linear system. Our results indicate that the proposed formulation provides sufficient accuracy. Furthermore, profiling of our prototype code shows that it is suitable for GPU acceleration, which may further improve its performance. The constraint preconditioner is shown to work well for cases with few, dense linear constraints, with a substantial improvement in linear solver convergence compared to the doubly augmented version. Our results indicate that our method can find solutions of acceptable accuracy in reasonable time for realistic problems from radiation therapy, as well as a simple test problem from support vector machine training. This is an extended version of a conference paper presented at the International Conference for Computational Science 2024 (Málaga, Spain) (Liu et al. 2024).