COAP 2004 Best Paper Award

Abstract

for their paper “Preconditioning Indefinite Systems in Interior Point Methods for Optimization”, published in Volume 28, pages 149–171. This paper describes a class of indefinite preconditioners for reduced KKT systems arising in quadratic and nonlinear optimization with interior point methods. Spectral analysis of preconditioners is given and the improvements resulting from the use of primal-dual regularization are demonstrated. Computational results are reported for the application of the preconditioner to a variety of medium-size convex quadratic programming problems. Work on the paper began during the Summer of 2001, about a year after the authors had first met when Zilli invited Gondzio to give a series of lectures on interior point methods in Padova in September, 2000. The authors had complementary research backgrounds: Bergamaschi and Zilli worked on the theory and implementation of iterative methods for linear and nonlinear systems of equations [2], while Gondzio worked on the design and implementation of interior point methods for large-scale optimization [3]. This was essential since together, the three authors could tackle in depth a problem which required expertice in several areas. At that time Bergamaschi and Zilli were impressed with the recent development of Lukysan and Vlycek [6], who applied conjugate gradients to indefinite systems, and with the later analysis of Keller, Gould and Wathen [4] of indefinite constraint preconditioners. They convinced Gondzio to look into the issue. In the meantime, Rozlozn´ok and Simoncini [7] came up with an improved understanding of issues connected with indefinite preconditioning (in the context of saddle point problems arising in partial differential equation); these developments led the authors to take a closer look at the problem. Starting in July, 2001, the authors began to incorporate into Gondzio’s interior point code HOPDM (higher order primal dual methods) four iterative techniques: conjugate gradients, BiCGstab, GMRES and QMR. A description of these schemes can be found in Kelley’s book [5] and the references therein. The authors realized that the key difficulty in the solution of augmented systems of Newton equations in interior point methods for quadratic and nonlinear programming (otherwise known as reduced KKT systems) consisted in the loss of sparsity caused by the presence of the Hessian matrix Q in the system � (Q + � 1