On the divergence of line search methods

Abstract

Abstract.We discuss the convergence of line search methods for minimization. We explain Mathematical subject classification: 20E28, 20G40, 20C20.

The divergence of the barycentric Padé interpolants

“…The fact that our algorithm to construct the function f (z) would not work in practice is an evidence of a deeper problem with this article and the theoretical analysis of the convergence of numerical algorithms in general. As we have shown in other instances [5,6,7,8,9], several algorithms which work well in practice are vulnerable to theoretical examples like the ones presented here. Therefore, the algorithm by Brezinski and Redivo-Zaglia may be quite adequate for all the degrees of the approximants one usually considers in practice (but this would be the subject of another article.…”

“…for V k in (4) and B k in (7). Thus,Ũ is non-singular and the determinant of the matrix N(ε) :=M+ εŨ is a polynomial Q(ε).…”

On numerical solving the spherical separability problem

“…At the same time, in nonconvex problems, the direct application of standard methods may have unpredictable consequences [13,15,16,20,21], and sometimes may even distract one from the desired solution. So, it seems to be quite natural (but hardly ever grounded) the reaction of the specialists propagating methods of direct selection-such as the method of branches and bounds (and cuts methods), which, as known, suffer the curse of dimension, when the volume of computations grows exponentially side by side with the growth of the problems dimension [13,15,16].…”

The divergence of the BFGS and Gauss Newton methods

“…with search directions d k ∈ R n and parameters α k ∈ R chosen with the intention that x k converge to a local minimizer of a function f : R n → R. Textbooks [2,14] present popular choices for the directions d k and the parameters α k and explain why they work in usual circumstances. The articles [4,5,12,13] analyze d k 's given by BFGS and conclude that these methods may not succeed in extreme situations. Our article [13] presents a similar result for the d k 's corresponding to Newton's method for minimization and hints that the techniques it describes could be also applied to other methods, but it does not elaborate on these possible extensions.…”

“…The articles [4,5,12,13] analyze d k 's given by BFGS and conclude that these methods may not succeed in extreme situations. Our article [13] presents a similar result for the d k 's corresponding to Newton's method for minimization and hints that the techniques it describes could be also applied to other methods, but it does not elaborate on these possible extensions. Finally, our article [11] also offers examples of unexpected behavior for Newton's method.…”