97,688 research outputs found
Product Form of the Inverse Revisited
Using the simplex method (SM) is one of the most effective ways of solving large scale real life linear optimization problems. The efficiency of the solver is crucial. The SM is an iterative procedure, where each iteration is defined by a basis of the constraint set. In order to speed up iterations, proper basis handling procedures must be applied.
Two methodologies exist in the state-of-the-art literature, the product form of the inverse (PFI) and lower-upper triangular (LU) factorization. Nowadays the LU method is widely used because 120-150 iterations can be done without the need of refactorization while the PFI can make only about 30-60 iterations without reinversion in order to maintain acceptable numerical accuracy.
In this paper we revisit the PFI and present a new version that can make hundreds or sometimes even few thousands of iterations without losing accuracy. The novelty of our approach is in the processing of the non-triangular part of the basis, based on block-triangularization algorithms. The new PFI performs much better than those found in the literature. The results can shed new light on the usefulness of the PFI
Isotropic functions revisited
To a smooth and symmetric function defined on a symmetric open set
and a real -dimensional vector space we
assign an associated operator function defined on an open subset
of linear transformations of , such that for
each inner product on , on the subspace
of -selfadjoint operators,
is the isotropic function associated to , which
means that , where denotes the
ordered -tuple of real eigenvalues of . We extend some well known
relations between the derivatives of and each to relations between
and . By means of an example we show that well known regularity
properties of do not carry over to .Comment: 13 pages. Added an example to show that loss of regularity is
possible. Extended the bibliography. Comments are welcom
The structure theory of set addition revisited
In this article we survey some of the recent developments in the structure
theory of set addition.Comment: 38p
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