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Quasidifferential Calculus pp 118–134Cite as

On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 29))

Abstract

The differential calculus for convex, compact-valued multifunctions developed by Tyurin, Banks and Jacobs is used to give an equivalent description in terms of multifunctions of the class of functions which can be represented as the difference of two globally Lipschitzian convex functions. This approach is also used to develop a means of representing piecewise-linear continuous functions as the difference of two piecewise-linear convex functions in finite dimensions. This leads directly to a Minkowski duality theorem for piecewise-linear positively homogeneous continuous functions and equivalence classes of convex compact sets produced by convex compact polyhedrons: every piecewise-linear positively homogeneous continuous function may be uniquely characterized by its quasidifferential (as defined by Demyanov and Rubinov) at zero.

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Authors and Affiliations

  1. Mathematics Division, Humboldt University, Berlin, GDR

    D. Melzer

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  1. D. Melzer
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V. F. Demyanov L. C. W. Dixon

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© 1986 The Mathematical Programming Society, Inc.

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Melzer, D. (1986). 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. (eds) Quasidifferential Calculus. Mathematical Programming Studies, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121142

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  • DOI: https://doi.org/10.1007/BFb0121142

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00928-0

  • Online ISBN: 978-3-642-00929-7

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