Local behavior of the Newton method on two equivalent systems from linear programming
Journal of Optimization Theory and Applications
Primal-dual Newton-type interior-point method for topology optimization
Journal of Optimization Theory and Applications
Inertia-controlling factorizations for optimization algorithms
Applied Numerical Mathematics
Nonlinear optimization and parallel computing
Parallel Computing - Special issue: Parallel computing in numerical optimization
Parallel Computing - Special issue: Parallel computing in numerical optimization
Simultaneous solution approaches for large optimization problems
Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
Newton-KKT interior-point methods for indefinite quadratic programming
Computational Optimization and Applications
Computational Optimization and Applications
Unified theory of augmented Lagrangian methods for constrained global optimization
Journal of Global Optimization
A stable primal---dual approach for linear programming under nondegeneracy assumptions
Computational Optimization and Applications
Computational Optimization and Applications
An aggregate deformation homotopy method for min-max-min problems with max-min constraints
Computational Optimization and Applications
On Interior-Point Warmstarts for Linear and Combinatorial Optimization
SIAM Journal on Optimization
Multilevel Algorithms for Large-Scale Interior Point Methods
SIAM Journal on Scientific Computing
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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.)