Matrix-Free Interior Point Method

Technical Report ERGO-09-012

J. Gondzio

In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.

Key words: Linear Programming, Quadratic Programming, Matrix-Free, Interior Point Methods, Iterative Methods, Implicit Preconditioner.

PDF ERGO-09-012.pdf.

Written: October 5, 2009, revised April 29, 2010.
Computational Optimization and Applications 51 (2012), No 2, pp 457-480.
DOI 10.1007/s10589-010-9361-3.

Related Software:
HOPDM Higher Order Primal Dual Method.