Piecewise linear secant approximation via algorithmic piecewise differentiation

@article{Griewank2018PiecewiseLS,
  title={Piecewise linear secant approximation via algorithmic piecewise differentiation},
  author={Andreas Griewank and Thomas Streubel and Lutz Lehmann and Manuel Radons and Richard Hasenfelder},
  journal={Optimization Methods and Software},
  year={2018},
  volume={33},
  pages={1108 - 1126}
}
It is shown how piecewise differentiable functions F:ℝn↦ℝm that are defined by evaluation programmes can be approximated locally by a piecewise linear model based on a pair of sample points . We show that the discrepancy between function and model at any point x is of the bilinear order . As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and… 
View on Taylor & Francis
arxiv.org
10 Citations
Methods Citations
3

10 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…
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.
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…
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.
Priority Programme 1962 Beyond the Oracle: Opportunities of Piecewise Differentiation
For more than thirty years much of the research and development in nonsmooth optimization has been predicated on the assumption that the user provides an oracle that evaluates at any given x ∈ R the…
On the Computational Complexity of the Chain Rule of Differential Calculus
TLDR
This work forms Chain Rule Differentiation and it is proved to be NP-complete, the fundamental prerequisite for computing accurate derivatives of composite functions which perform a potentially very large number of elemental function evaluations.
Beyond the Oracle: Opportunities of Piecewise Differentiation
For more than 30 years much of the research and development in nonsmooth optimization has been predicated on the assumption that the user provides an oracle that evaluates at any given \({\boldsymbol…
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.
Convergence results for some piecewise linear solvers
TLDR
It is shown that the piecewise linear equation system, based on the jats:inline-formula, can be implemented as a discrete-time solution to the inequality problem.
Feature Representation and Similarity Measure Based on Covariance Sequence for Multivariate Time Series
TLDR
The experimental results demonstrate that the proposed method not only can handle multivariate time series with different lengths but also is more efficient and effective than the existing methods for the MTS data mining.

References

SHOWING 1-10 OF 21 REFERENCES
On stable piecewise linearization and generalized algorithmic differentiation
It is shown how functions that are defined by evaluation programs involving the absolute value function abs() (besides smooth elementals) can be approximated locally by piecewise-linear models in the…
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.
Solving piecewise linear systems in abs-normal form
Direct solution of piecewise linear systems
  • Manuel Radons
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 2016
Introduction to Piecewise Differentiable Equations
-1. Sample problems for nonsmooth equations. -2. Piecewise affline functions. -3. Elements from nonsmooth analysis. -4. Piecewise differentiable functions. -5. Sample applications.
Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition
TLDR
This second edition has been updated and expanded to cover recent developments in applications and theory, including an elegant NP completeness argument by Uwe Naumann and a brief introduction to scarcity, a generalization of sparsity.
Iterative solution of nonlinear equations in several variables
TLDR
Convergence of Minimization Methods An Annotated List of Basic Reference Books Bibliography Author Index Subject Index.
Representation and Analysis of Piecewise Linear Functions in Abs-Normal Form
TLDR
The system of equations \(F(x) = 0\) can be rewritten in terms of the variable vector \(z\) as a linear complementarity problem (LCP) and the transformation itself and the properties of the LCP depend on the Schur complement.
NEWTON's METHOD FOR NON-DIFFERENTIABLE FUNCTIONS
  • B. Kummer
  • Mathematics
    Advances in Mathematical Optimization
  • 1988
On propose des extensions de la methode de Newton pour resoudre des equations avec des fonctions non differentiables. On donne des conditions de convergence, des estimations de la vitesse ainsi que…
Introduction to Interval Computation
...
1
2
3
...