Ill-Conditioning and Computational Error in Interior Methods for Nonlinear Programming

Ill-conditioning has long been regarded as a plague on interior methods, but its damaging effects have rarely been documented. In fact, implementors of interior methods who ignore warnings about the dire consequences of ill-conditioning usually manage to compute accurate solutions. We offer some insight into this seeming contradiction by analyzing ill-conditioning within a primal-dual method in which the full, usually well-conditioned primal-dual matrix is transformed to a "condensed," inherently ill-conditioned matrix Mpd . We show that ill-conditioning in the exact condensed matrix closely resembles that known for the primal barrier Hessian, and then examine the influence of cancellation in the computed constraints. Using the structure of Mpd , various bounds are obtained on the absolute accuracy of the computed primal-dual steps. Without cancellation, the portion of the computed x step in the small space of Mpd (a subspace close to the null space of the Jacobian of the active constraints) has an absolute error bound comparable to machine precision, and its large-space component has a much smaller error bound. With cancellation (the usual case), the absolute error bounds for both the small- and large-space components of the computed x step are comparable to machine precision. In either case, the absolute error bound for the computed multiplier steps associated with active constraints is comparable to machine precision; the computed multiplier steps for inactive constraints, although converging to zero, retain (approximately) full relative precision. Because of errors in forming the right-hand side, the absolute error in the computed solution of the full, well-conditioned primal-dual system is shown to be comparable to machine precision. Thus, under quite general conditions, ill-conditioning in Mpd does not noticeably impair the accuracy of the computed primal-dual steps. (A similar analysis applies to search directions obtained by direct solution of the primal Newton equations.)