M. H. Wright, Some properties of the Hessian of the logarithmic barrier function, Math. Programming, 67(1994), pp. 265-295. 21  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the log barrier is a particular instance. Following the results of Murray [16] and Lootsma [14] regarding the ill conditioning of the Hessian matrix P xx ( Delta; along the central path, the nature of the ill conditioning in the neighborhood of the solution is examined further by M. H. Wright [23]. The latter paper proposes techniques for calculating approximate Newton steps for the function P ( Delta; that do not require the solution of ill conditioned systems. In earlier work, Gould [12] proposed a method for computing accurate Newton steps by identifying the active indices explicitly, ....

....computing accurate Newton steps by identifying the active indices explicitly, and forming an augmented linear system that remains well conditioned even when is small. The effect of finite precision arithmetic on the calculation of Newton steps is examined by M. H. Wright [24] Both M. H. Wright [23,24] and S. J. Wright [26] use a subspace decomposition of the Hessian P xx ( Delta; like the one used in Section 3 below, but there is an important distinction that we note later. The paper [26] addresses the issue of domain of convergence of Newton s method applied to P ( Delta; which is also ....

[Article contains additional citation context not shown here]

M. H. Wright. Some properties of the Hessian of the logarithmic barrier function. Mathematical Programming, 67:265--295, 1994.

....by a symmetric row and column exchange. The procedure terminates when none of the remaining diagonal elements is large enough to qualify as a pivot, and an approximate solution is computed with the partial factors. Higham [7, Chapter 10] presents an error analysis of this approach, and M. Wright [15] has considered its use in factoring the Hessian matrices that arise in the Newton logarithmic barrier method for nonlinear programming. This strategy is not practical in the context of interior point linear programming codes because the matrices in question are too large to allow row and column ....

....section applies to primal and dual degenerate linear programs. We conclude with some computational results in Section 6. A number of other theoretical papers on linear algebra operations in barrier and interior point methods have appeared in recent years. We mentioned above the paper of M. Wright [15], in which a Cholesky procedure with diagonal pivoting is used as the basis of an algorithm to construct steps that are accurate both in the subspace spanned by the active constraint Jacobian and its complement. Our focus in the current paper is on (possibly degenerate) linear programs rather than ....

M. H. Wright, Some properties of the Hessian of the logarithmic barrier function, Mathematical Programming, 67 (1994), pp. 265--295.

....not yet understood, but perhaps one day they will be. 4.3. Barrier methods revisited The problem of ill conditioning, as noted earlier, has haunted interior methods since the late 1960s, but there has been substantial recent progress in understanding this issue. A detailed analysis was given in [34] of the structure of the primal barrier Hessian (2.14) in an 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 ....

M. H. Wright (1994). Some properties of the Hessian of the logarithmic barrier function, Math. Prog. 67, 265--295.

....P ( Delta; shrinks as # 0, but the rate of shrinkage is not especially severe. Key to our analysis is a partitioning of the space IR n into the range space of the active constraint Jacobian and its complement. A decomposition of this type has been used previously (for example, by M. Wright [25 27]) to analyze the properties of the gradient and Hessian of P ( Delta; we mention some specific connections below. We assume a priori that w lies in the neighborhood kw Gamma x( k C oe ; 19) where C 0 and oe 1 are given constants, and that 2 (0; 20) for some 0. We show later ....

....i O( min(2;oe) Theta( i = 1; 2; q; 23) for all sufficiently small. Using this observation, we examine the relevant properties of P xx (w; The nature of the ill conditioning in this matrix at minimizing points w = x( was examined by Lootsma [18] and Murray [19] M. Wright [25] further examined the eigenstructure of this matrix in a strictly feasible neighborhood of the solution x , for small , and essentially showed in [25, Lemmas 3.1, 3.2, Theorem 3.1] that this matrix has q eigenvalues of size Theta( Gamma1 ) and n Gamma q eigenvalues of size O(1) Drawing on ....

[Article contains additional citation context not shown here]

M. H. Wright. Some properties of the Hessian of the logarithmic barrier function. Mathematical Programming, 67:265--295, 1994.

....the log barrier is a particular instance. Following the results of Murray [16] and Lootsma [14] regarding the ill conditioning of the Hessian matrix P xx ( Delta; along the central path, the nature of the ill conditioning in the neighborhood of the solution is examined further by M. H. Wright [23]. The latter paper proposes techniques for calculating approximate Newton steps for the function P ( Delta; that do not require the solution of ill conditioned systems. In earlier work, Gould [12] proposed a method for computing accurate Newton steps by identifying the active indices ....

....computing accurate Newton steps by identifying the active indices explicitly, and forming an augmented linear system that remains well conditioned even when is small. The effect of finite precision arithmetic on the calculation of Newton steps is examined by M. H. Wright [24] Both M. H. Wright [23,24] and S. J. Wright [26] use a subspace decomposition of the Hessian P xx ( Delta; like the one used in Section 3 below, but there is an important distinction that we note later. The paper [26] addresses the issue of domain of convergence of Newton s method applied to P ( Delta; which is ....

[Article contains additional citation context not shown here]

M. H. Wright. Some properties of the Hessian of the logarithmic barrier function. Mathematical Programming, 67:265--295, 1994.

....of the Log Barrier Function on Degenerate Nonlinear Programs 3 of Murray [16] and Lootsma [14] regarding the ill conditioning of the Hessian matrix P xx ( Delta; along the central path, the nature of the ill conditioning in the neighborhood of the solution is examined further by M. H. Wright [23]. The latter paper proposes techniques for calculating approximate Newton steps for the function P ( Delta; that do not require the solution of ill conditioned systems. In earlier work, Gould [12] proposed a method for computing accurate Newton steps by identifying the active indices ....

....computing accurate Newton steps by identifying the active indices explicitly, and forming an augmented linear system that remains well conditioned even when is small. The effect of finite precision arithmetic on the calculation of Newton steps is examined by M. H. Wright [24] Both M. H. Wright [23,24] and S. J. Wright [26] use a subspace decomposition of the Hessian P xx ( Delta; like the one used in Section 3 below, but there is an important distinction that we note later. The paper [26] addresses the issue of domain of convergence of Newton s method applied to P ( Delta; which is ....

[Article contains additional citation context not shown here]

M. H. Wright. Some properties of the Hessian of the logarithmic barrier function. Mathematical Programming, 67:265--295, 1994.

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M. H. Wright, Some properties of the Hessian of the logarithmic barrier function, Math. Programming, 67(1994), pp. 265-295. 21

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WRIGHT, M.H., Some Properties of the Hessian in the Logarithmic Barrier Function, Mathematical Programming, Vol. 67, pp.265-295, 1994.

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