A novel class of approximate inverse preconditioners for large positive definite linear systems in optimization

We propose a class of preconditioners for large positive definite linear systems, arising in nonlinear optimization frameworks. These preconditioners can be computed as by-product of Krylov-subspace solvers. Preconditioners in our class are chosen by setting the values of some user-dependent parameters. We first provide some basic spectral properties which motivate a theoretical interest for the proposed class of preconditioners. Then, we report the results of a comparative numerical experience, among some preconditioners in our class, the unpreconditioned case and the preconditioner in Fasano and Roma (Comput Optim Appl 56:253–290, 2013). The experience was carried on first considering some relevant linear systems proposed in the literature. Then, we embedded our preconditioners within a linesearch-based Truncated Newton method, where sequences of linear systems (namely Newton’s equations), are required to be solved. We performed an extensive numerical testing over the entire medium-large scale convex unconstrained optimization test set of CUTEst collection (Gould et al. Comput Optim Appl 60:545–557, 2015), confirming the efficiency of our proposal and the improvement with respect to the preconditioner in Fasano and Roma (Comput Optim Appl 56:253–290, 2013).