Quadratic reformulations of nonlinear binary optimization problems

Martin Anthony, Endre Boros, Yves Crama, Aritanan Gruber

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all “minimal” quadratizations of negative monomials.

Original languageEnglish (US)
Pages (from-to)115-144
Number of pages30
JournalMathematical Programming
Issue number1-2
StatePublished - Mar 1 2017

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)


  • Nonlinear binary optimization
  • Pseudo-Boolean functions
  • Quadratic binary optimization
  • Reformulation methods

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