S. J. Wright (1998). Finite-precision effects on the local convergence of interior-point algorithms for nonlinear programming, Preprint ANL/MCS P705-0198, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois. Home/Search Document Not in Database Summary Related Articles Check |
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Active Set and Interior Methods for Nonlinear Optimization - Byrd, Nocedal (Correct)
....of order O(1= This is reflected in the primal dual iteration (12) where the matrix Sigma = S Gamma1 becomes unbounded as 0. Nevertheless, solving (12) by a direct method, as is done in most linear programming codes, does not lead to significant roundoff errors, even when is very small [11, 12]. The key observation in this roundoff error analysis can be better explained if we consider Newton like methods for solving the unconstrained problem min f(x) Here the step p is computed by solving a system of the form Ap = Gammarf (x) where A is either the Hessian matrix r 2 f(x) or some ....
....to changes in A. The ill conditioning of the barrier function can cause errors in the factorization of the iteration matrix, but very significant errors can be tolerated before the quality of the iteration is degraded and simple safeguards ensure that high accuracy is obtained in most cases [12]. All of this assumes that a direct method is used to solve (12) But in many practical applications, the problem is so large that direct methods are impractical due to the great amount of fill that occurs in the factorization. In other applications, the Hessians of f; g or h are not be ....
S.J. Wright. Finite-Precision Effects on the Local Convergence of InteriorPoint Algorithms for Nonlinear Programming, Preprint ANL/MCS P7050198, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois.
The Interior-Point Revolution in Constrained Optimization - Wright (1998) (5 citations) Self-citation (Wright) (Correct)
....entire neighborhood of the solution, along with a discussion of techniques for finessing the ill conditioning. Several papers ( 12] 10] 37] 39] 40] have analyzed the stability of specific factorizations for various interior methods. Very recently, the surprising result was obtained ( 36] [41]) that, under conditions that normally hold in practice, ill conditioning of certain key matrices in interior methods does not noticeably degrade the accuracy of the computed search directions. In particular, if a backward stable method is used to solve the condensed primal dual system (4.7) the ....
S. J. Wright (1998). Finite-precision effects on the local convergence of interior-point algorithms for nonlinear programming, Preprint ANL/MCS P705-0198, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois.
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