Mathematical Programming

, Volume 162, Issue 1–2, pp 225–240 | Cite as

Mixed-integer quadratic programming is in NP

  • Alberto Del Pia
  • Santanu S. Dey
  • Marco Molinaro
Full Length Paper Series A


Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73–77 [17]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299–304 [1], von zur Gathen and Sieveking in Proc Am Math Soc 72:155–158 [18], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161–172. Springer [9], Papadimitriou in J Assoc Comput Mach 28:765–768 [15]).


Quadratic programming Integer programming Complexity 

Mathematics Subject Classification

90C11 90C20 90C60 


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Alberto Del Pia
    • 1
  • Santanu S. Dey
    • 2
  • Marco Molinaro
    • 2
  1. 1.Department of Industrial and Systems Engineering & Wisconsin Institute for DiscoveryUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Computer Science DepartmentPUC-RioRio de JaneiroBrazil

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