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Piecewise linear secant approximation via Algorithmic Piecewise Differentiation

Please always quote using this URN: urn:nbn:de:0297-zib-61642
  • It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. 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 convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

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Author:Andreas GriewankORCiD, Tom StreubelORCiD, Lutz Lehmann, Richard HasenfelderORCiD, Manuel RadonsORCiD
Document Type:ZIB-Report
Tag:ADOL-C; Automatic differentiation; Computational graph; Generalized Hermite interpolation; Lipschitz continuity
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area)
CCS-Classification:G. Mathematics of Computing
Date of first Publication:2016/12/28
Series (Serial Number):ZIB-Report (16-54)
Published in:published at Optimization Methods and Software