Direct solution of piecewise linear systems

@article{Radons2016DirectSO,
  title={Direct solution of piecewise linear systems},
  author={Manuel Radons},
  journal={Theor. Comput. Sci.},
  year={2016},
  volume={626},
  pages={97-109}
}
  • Manuel Radons
  • Published 2 May 2016
  • Mathematics, Computer Science
  • Theor. Comput. Sci.
View PDF on arXiv
12 Citations
Background Citations
4
Methods Citations
3

12 Citations

Citation Type

Citation Type

An Open Newton Method for Piecewise Smooth Functions
Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth…
Solving piecewise linear systems in abs-normal form
Sign controlled solvers for the absolute value equation with an application to support vector machines
TLDR
Three solvers are presented: One direct, one semi-iterative and one discrete variant of damped Newton, and their previously proved ranges of correctness and convergence, respectively, are extended.
(Almost) matrix‐free solver for piecewise linear functions in abs‐normal form
  • T. Bosse
  • Computer Science, Mathematics
    Numer. Linear Algebra Appl.
  • 2019
TLDR
The first (almost) matrix‐free versions of some solver for ANFs are presented and the question if a solver is based on the ANF and uses the (Schur‐complement) matrices of the explicit ANF representation, it has to be considered computationally expensive is addressed.
On the abs-polynomial expansion of piecewise smooth functions
TLDR
It is shown that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case and extended to infinite series that converge absolutely on spherical domains.
Piecewise Polynomial Taylor Expansions—The Generalization of Faà di Bruno’s Formula
We present an extension of Taylor’s theorem towards nonsmooth evaluation procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in…
Piecewise linear secant approximation via algorithmic piecewise differentiation
TLDR
A generalized Newton's method based on successive piecewise linearization is devised and sufficient conditions for convergence and convergence rates equalling those of semismooth Newton are proved.
Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation
TLDR
A generalized trapezoidal rule is proposed for initial value problems with piecewise smooth right-hand side based on a generalization of algorithmic differentiation that can achieve a higher convergence order than with the classical method.
Accelerating the Lawson-Hanson NNLS solver for large-scale Tchakaloff regression designs
TLDR
This work cope the problem of computing near G-optimal compressed designs for high-degree polynomial regression on fine discretizations of 2d and 3d regions of arbitrary shape via an improved version of the Lawson-Hanson NNLS solver for the corresponding full and largescale underdetermined moment system.
Deviation Maximization for Rank-Revealing QR Factorizations
In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block…
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References

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Solving piecewise linear systems in abs-normal form
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Iterative Solution of Piecewise Linear Systems
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In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps.
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This work investigates equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax+B|x| = b and shows that this absolute value equation is NP-hard to solve, and that solving it with B = I solves the general linear complementarity problem.
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A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations.
Absolute value equations
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