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Mixed-integer quadratic programming is in NP

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Abstract

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]).

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering & Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, USA

    Alberto Del Pia

  2. Computer Science Department, PUC-Rio, Rio de Janeiro, Brazil

    Santanu S. Dey & Marco Molinaro

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  1. Alberto Del Pia
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  2. Santanu S. Dey
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  3. Marco Molinaro
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Correspondence to Marco Molinaro.

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Pia, A.D., Dey, S.S. & Molinaro, M. Mixed-integer quadratic programming is in NP. Math. Program. 162, 225–240 (2017). https://doi.org/10.1007/s10107-016-1036-0

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  • DOI: https://doi.org/10.1007/s10107-016-1036-0

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