Representations of continuous piecewise affine functions

Authors

  • Vassili N. Malozemov St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation
  • Grigoriy Sh. Tamasyan St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation; Institute of Problems of Mechanical Engineering of the Russian Academy of Sciences, 61, Bolshoy pr. V.O., St Petersburg, 199178, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2022.106

Abstract

Continuous piecewise affine functions are widely used in computational mathematics. In one-dimensional case such functions are called broken lines. The article analyzes the analytical representations of broken lines in the forms accepted in the theory of polynomial splines as well as in the form of the difference of the maxima of two finite families of affine functions. The connection between these representation is being determined.

Keywords:

piecewise affine function, broken line, analytical representations of broken lines, difference of convex functions

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References

Литература

1. Малозёмов В.Н., Певный А.Б. Полиномиальные сплайны. Ленинград, Изд-во Ленингр. ун-та (1986).

2. Melzer D. On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions. In: Demyanov V.F., Dixon L.C.W. (ed.) Quasidifferential Calculus, 118–134. Berlin, Heidelberg, Springer (1986).

3. Kripfganz A., Schulze R. Piecewise affine functions as a difference of two convex functions. Optimization 18 (1), 23–29 (1987). https://doi.org/10.1080/02331938708843210

4. Gorokhovik V.V., Zorko O. I. Piecewise affine functions and polyhedral sets. Optimization 31 (3), 209–221 (1994). https://doi.org/10.1080/02331939408844018

References

1. Malozemov V.N., Pevnyj A.B. Polynomial splines. Leningrad, Leningrad University Press (1986). (In Russian)

2. Melzer D. On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions. In: Demyanov V.F., Dixon L.C.W. (ed.) Quasidifferential Calculus, 118–134. Berlin, Heidelberg, Springer (1986).

3. Kripfganz A., Schulze R. Piecewise affine functions as a difference of two convex functions. Optimization 18 (1), 23–29 (1987). https://doi.org/10.1080/02331938708843210

4. Gorokhovik V.V., Zorko O. I. Piecewise affine functions and polyhedral sets. Optimization 31 (3), 209–221 (1994). https://doi.org/10.1080/02331939408844018

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Published

2022-04-10

How to Cite

Malozemov, V. N., & Tamasyan, G. S. (2022). Representations of continuous piecewise affine functions. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(1), 53–63. https://doi.org/10.21638/spbu01.2022.106

Issue

Vol. 9 No. 1 (2022)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.