# Solving piecewise linear systems in abs-normal form

@article{Griewank2015SolvingPL, title={Solving piecewise linear systems in abs-normal form}, author={Andreas Griewank and Jens Uwe Bernt and Manuel Radons and Thomas Streubel}, journal={Linear Algebra and its Applications}, year={2015}, volume={471}, pages={500-530} }

## 31 Citations

An algorithm for nonsmooth optimization by successive piecewise linearization

- MathematicsMath. Program.
- 2019

It is shown that the new method when started from within a compact level set generates a sequence of iterates whose cluster points are all Clarke stationary, which illustrates the capabilities of the proposed approach.

Finite convergence of an active signature method to local minima of piecewise linear functions

- Computer ScienceOptim. Methods Softw.
- 2019

A new method is presented that computes a local minimizer of the proximal model objective, which is also known as criticality in nonsmooth optimization, and provides opportunities for structure exploitation like warm starts in the context of the nonlinear, outer loop.

(Almost) matrix‐free solver for piecewise linear functions in abs‐normal form

- Computer Science, MathematicsNumer. Linear Algebra Appl.
- 2019

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.

Algorithmic differentiation for piecewise smooth functions: a case study for robust optimization

- Computer ScienceOptim. Methods Softw.
- 2018

This paper presents a minimization method for Lipschitz continuous, piecewise smooth objective functions based on algorithmic differentiation (AD), and presents corresponding drivers for the AD tool ADOL-C which are embedded in the nonsmooth solver LiPsMin.

On Lipschitz optimization based on gray-box piecewise linearization

- MathematicsMath. Program.
- 2016

This work addresses the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function and demonstrates how the local model can be minimized by a bundle-type method, which benefits from the availability of additional gray-box information via the abs-normal form.

Enumeration of subdifferentials of piecewise linear functions with abs-normal form

- Mathematics, Computer ScienceOptim. Methods Softw.
- 2018

A method for computing and enumerating the elements of the limiting subdifferential at a given non-differential point is described with branch and bound search and it is shown that there may be some cases in which computational work may be reduced using the branch and Bound search.

An Open Newton Method for Piecewise Smooth Functions

- Mathematics
- 2018

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…

First- and second-order optimality conditions for piecewise smooth objective functions

- Mathematics, Computer ScienceOptim. Methods Softw.
- 2016

This work describes local optimality by first- and second-order necessary and sufficient conditions, which generalize the corresponding Kuhn-Tucker-Karush (KKT) theory for smooth problems and exemplifies the theory on two nonsmooth examples of Nesterov.

Direct solution of piecewise linear systems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2016

Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

- MathematicsOptim. Methods Softw.
- 2018

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.

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