Mathematical Programming

, Volume 151, Issue 1, pp 191–223 | Cite as

On mathematical programming with indicator constraints

  • Pierre Bonami
  • Andrea Lodi
  • Andrea Tramontani
  • Sven Wiese
Full Length Paper Series B


In this paper we review the relevant literature on mathematical optimization with logical implications, i.e., where constraints can be either active or disabled depending on logical conditions to hold. In the case of convex functions, the theory of disjunctive programming allows one to formulate these logical implications as convex nonlinear programming problems in a space of variables lifted with respect to its original dimension. We concentrate on the attempt of avoiding the issue of dealing with large NLPs. In particular, we review some existing results that allow to work in the original space of variables for two relevant special cases where the disjunctions corresponding to the logical implications have two terms. Then, we significantly extend these special cases in two different directions, one involving more general convex sets and the other with disjunctions involving three terms. Computational experiments comparing disjunctive programming formulations in the original space of variables with straightforward bigM ones show that the former are computationally viable and promising.


Disjunctive programming bigM method Perspective reformulation On/off constraints  

Mathematics Subject Classification

Primary 90C11 Secondary 90C25 90C57 



The second author acknowledges the support of MIUR, Italy, under the grant PRIN 2012. The fourth author acknowledges the support of the EU ITN 316647 “Mixed-Integer Nonlinear Optimization” (MINO). We are indebted to four anonymous referees for a careful reading and insightful suggestions.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Pierre Bonami
    • 1
  • Andrea Lodi
    • 2
  • Andrea Tramontani
    • 3
  • Sven Wiese
    • 2
  1. 1.CPLEX OptimizationIBM SpainMadridSpain
  2. 2.Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione “Guglielmo Marconi”Università di BolognaBolognaItaly
  3. 3.CPLEX OptimizationIBM ItalyBolognaItaly

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