R.M. Gower and J. Gondzio
At the heart of Newton based optimization methods is a sequence of symmetric linear systems. Each consecutive system in this sequence is similar to the next, so solving them separately is a waste of computational effort. Here we describe automatic preconditioning techniques for iterative methods for solving such sequences of systems by maintaining an estimate of the inverse system matrix. We update the estimate of the inverse system matrix with quasi-Newton type formulas based on what we call an action constraint instead of the secant equation. We implement the estimated inverses as preconditioners in a Newton-CG method and prove quadratic termination. Our implementation is the first parallel quasi-Newton preconditioners, in full and limited memory variants. Tests on logistic Support Vector Machine problems reveal that our method is very efficient, converging in wall clock time before a Newton-CG method without preconditioning. Further tests on a set of classic test problems reveal that the method is robust. The action constraint makes these updates flexible enough to mesh with trust-region and active set methods, a flexibility that is not present in classic quasi-Newton methods.
Key words: quasi-Newton method, inexact Newton method, preconditioners, linear systems, conjugate gradients, balancing preconditioner.