  The low-rank alternating directions implicit (LR-ADI) iteration is a
frequently employed method for efficiently computing low-rank approximate
solutions of large-scale Lyapunov equations. In order to achieve a rapid error
reduction, the iteration requires shift parameters whose selection and
generation is often a difficult task, especially for nonsymmetric coefficients
in the Lyapunov equation. This article represents a follow up of Benner et al.
[ETNA, 43 (2014-2015), pp. 142-162] and investigates self-generating shift
parameters based on a minimization principle for the Lyapunov residual norm.
Since the involved objective functions are too expensive to evaluate and,
hence, intractable, compressed objective functions are introduced which are
efficiently constructed from the available data generated by the LR-ADI
iteration. Several numerical experiments indicate that these residual
minimizing shifts using approximated objective functions outperform existing
precomputed and dynamic shift parameter selection techniques, although their
generation is more involved.
