Mixed-Integer Convex Representability

  • Miles Lubin
  • Ilias Zadik
  • Juan Pablo VielmaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.


Natural Number Finite Union Recession Cone Rational Polyhedron Rational Subspace 
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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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