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Perturbation of error bounds
Our aim in the current article is to extend the developments in Kruger, Ngai & Théra , SIAM J. Optim. 20(6), 3280–3296 (2010) and, more precisely, to characterize , in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the 'radius of error bounds'. The definitions and characterizations are illustrated by examples
The error and perturbation bounds for the absolute value equations with some applications
To our knowledge, so far, the error and perturbation bounds for the general
absolute value equations are not discussed. In order to fill in this study gap,
in this paper, by introducing a class of absolute value functions, we study the
error bounds and perturbation bounds for two types of absolute value equations
(AVEs): Ax-B|x|=b and Ax-|Bx|=b. Some useful error bounds and perturbation
bounds for the above two types of absolute value equations are presented. By
applying the absolute value equations, we also obtain the error and
perturbation bounds for the horizontal linear complementarity problem (HLCP).
In addition, a new perturbation bound for the LCP without constraint conditions
is given as well, which are weaker than the presented work in [SIAM J. Optim.,
2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some
computable estimates for the above upper bounds are given, which are sharper
than some existing results under certain conditions. Some numerical examples
for the AVEs from the LCP are given to show the feasibility of the perturbation
bounds
Sharp entrywise perturbation bounds for Markov chains
For many Markov chains of practical interest, the invariant distribution is
extremely sensitive to perturbations of some entries of the transition matrix,
but insensitive to others; we give an example of such a chain, motivated by a
problem in computational statistical physics. We have derived perturbation
bounds on the relative error of the invariant distribution that reveal these
variations in sensitivity.
Our bounds are sharp, we do not impose any structural assumptions on the
transition matrix or on the perturbation, and computing the bounds has the same
complexity as computing the invariant distribution or computing other bounds in
the literature. Moreover, our bounds have a simple interpretation in terms of
hitting times, which can be used to draw intuitive but rigorous conclusions
about the sensitivity of a chain to various types of perturbations
Error analysis of householder transformations as applied to the standard and generalized eigenvalue problems
Backward error analyses of the application of Householder transformations to both the standard and the generalized eigenvalue problems are presented. The analysis for the standard eigenvalue problem determines the error from the application of an exact similarity transformation, and the analysis for the generalized eigenvalue problem determines the error from the application of an exact equivalence transformation. Bounds for the norms of the resulting perturbation matrices are presented and compared with existing bounds when known
Perturbation of error bounds
Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi: 10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples.The research is supported by the Australian Research Council: project DP160100854; EDF and the Jacques Hadamard Mathematical Foundation: Gaspard Monge Program for Optimization and Operations Research. The research of the second and third authors is also supported by MINECO of Spain and FEDER of EU: Grant MTM2014-59179-C2-1-P
Derivatives of Multilinear Functions of Matrices
Perturbation or error bounds of functions have been of great interest for a
long time. If the functions are differentiable, then the mean value theorem and
Taylor's theorem come handy for this purpose. While the former is useful in
estimating in terms of and requires the norms of the
first derivative of the function, the latter is useful in computing higher
order perturbation bounds and needs norms of the higher order derivatives of
the function.
In the study of matrices, determinant is an important function. Other scalar
valued functions like eigenvalues and coefficients of characteristic polynomial
are also well studied. Another interesting function of this category is the
permanent, which is an analogue of the determinant in matrix theory. More
generally, there are operator valued functions like tensor powers,
antisymmetric tensor powers and symmetric tensor powers which have gained
importance in the past. In this article, we give a survey of the recent work on
the higher order derivatives of these functions and their norms. Using Taylor's
theorem, higher order perturbation bounds are obtained. Some of these results
are very recent and their detailed proofs will appear elsewhere.Comment: 17 page
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