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Interior point methods are widely used for different types of mathematicaloptimization problems. Many implementations of interior point methods in usetoday rely on direct linear solvers to solve systems of equations in eachiteration. The need to solve ever larger optimization problems more efficientlyand the rise of hardware accelerators for general purpose computing has led toa large interest in using iterative linear solvers instead, with the majorissue being inevitable ill-conditioning of the linear systems arising as theoptimization progresses. We investigate the use of Krylov solvers for interiorpoint methods in solving optimization problems from radiation therapy. Weimplement a prototype interior point method using a so called doubly augmentedformulation of the Karush-Kuhn-Tucker (KKT) linear system of equations,originally proposed by Forsgren and Gill, and evaluate its performance on realoptimization problems from radiation therapy. Crucially, our implementationuses a preconditioned conjugate gradient method with Jacobi preconditioninginternally. Our measurements of the conditioning of the linear systems indicatethat the Jacobi preconditioner improves the conditioning of the systems to adegree that they can be solved iteratively, but there is room for furtherimprovement in that regard. Furthermore, profiling of our prototype code showsthat it is suitable for GPU acceleration, which may further improve itsperformance in practice. Overall, our results indicate that our method can findsolutions of acceptable accuracy in reasonable time, even with a simple Jacobipreconditioner.

Krylov Solvers for Interior Point Methods with Applications in Radiation Therapy