Optimization Letters

, Volume 13, Issue 2, pp 439–443 | Cite as

A note on surjectivity of piecewise affine mappings

  • Manuel RadonsEmail author
Short Communication


A standard theorem in nonsmooth analysis states that a piecewise affine function \(F:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero determinant sign. In this note we prove that surjectivity already follows from coherent orientation of the selection functions which are active on the unbounded sets of a polyhedral subdivision of the domain corresponding to F. A side bonus of the argumentation is a short proof of the classical statement that an injective piecewise affine function is coherently oriented.


Nonsmooth optimization Piecewise affine functions Piecewise linear functions Surjectivity criteria 


  1. 1.
    Fusek, P.: On metric regularity for weakly almost piecewise smooth functions and some applications in nonlinear semidefinite programming. SIAM J. Optim. 23(2), 1041–1061 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Outerelo, E., Ruiz, J.M.: Mapping Degree Theory. American Mathematical Society, Providence (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ziegler, G.M.: Lectures on Polytopes. Springer, Berlin (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Workgroup Discrete Mathematics/GeometryTechnical University of BerlinBerlinGermany

Personalised recommendations