Optimization by Direct Search in Matrix Computations

Abstract

of algorithms it is shown that direct search is capable of revealing instability or poor performance, even when such failure is difficult to discover using theoretical analysis or numerical tests with random or nonrandom data. Informative numerical examples generated by direct search provide the impetus for further analysis and improvement of an algorithm. The direct search methods used are the method of alternating directions and the multi-directional search method of Dennis and Torczon. The problems examined include the reliability of matrix condition number estimators and the stability of Strassen's fast matrix inversion method.

Altered effective connectivity and anomalous anatomy in the basal ganglia-thalamocortical circuit of stuttering speakers

“…LISREL 8.7 was used to run SEM. It used an iterative maximum likelihood algorithm to calculate path coefficients and to achieve the best match between the covariance matrix reproduced by the model and the observed variance-covariance structure from the data (Higham, 1993;Jö reskog and Sorbom, 1996). Derived from the logarithmic expression of a likelihood ratio test, the maximum likelihood (ML) discrepancy function was introduced Jö reskog, 1967) to indicate the fit of the model.…”

Analysis of direct searches for discontinuous functions

“…Another example arises in the tuning of algorithmic parameters for a given method/code (see [6] for instances where DSM have been applied to solve such problems) -the resulting objective functions are likely to exhibit all sorts of discontinuities given the way that typically a method/code responds to changes in its parameters. DSM have also been used for automatic error analysis [12,13], a process in which the computer is used to analyze the accuracy or stability of a numerical computation (and examples have been provided where the objective function is discontinuous). Many engineering design problems (which are likely to form the core of the mainstream applications of derivative-free optimization) lead to objective functions involving discontinuities and limited or no access to derivatives (one such application in aircraft design which was recently drawn to our attention is reported in [3]).…”

Testing linear algebra software

“…In some experiments on direct search in Higham (1993) we used Matlab 3.5, and we found it much easier to generate counterexamples to rcond (examples in which rcond provides an estimate that is much too large) than we do now with Matlab 4.2. It seems that the maximizations in Higham (1993) were not only defeating the algorithm underlying rcond, but also, unbeknown to us, exploiting a bug in the implementation of the function. The conclusions of Higham (1993) are unaffected, however.…”

“…It seems that the maximizations in Higham (1993) were not only defeating the algorithm underlying rcond, but also, unbeknown to us, exploiting a bug in the implementation of the function. The conclusions of Higham (1993) are unaffected, however.…”

“…The other is to attempt to maximize the function in terms of the problem data, using an optimization routine. This latter technique is investigated in Higham (1993) (see also Higham (1996, Ch. 24)), where direct search methods are shown to be appropriate.…”