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There Cannot be any Algorithm for Integer Programming with Quadratic Constraints

Author

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  • R. C. Jeroslow

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

Abstract

This paper studies a class of integer programming problems in which squares of variables may occur in the constraints, and shows that no computing device can be programmed to compute the optimum criterion value for all problems in this class.

Suggested Citation

  • R. C. Jeroslow, 1973. "There Cannot be any Algorithm for Integer Programming with Quadratic Constraints," Operations Research, INFORMS, vol. 21(1), pages 221-224, February.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:1:p:221-224
    DOI: 10.1287/opre.21.1.221
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    Cited by:

    1. Hamidur Rahman & Ashutosh Mahajan, 2020. "On the facet defining inequalities of the mixed-integer bilinear covering set," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 545-575, December.
    2. Hoai Le Thi & Duc Tran, 2014. "Optimizing a multi-stage production/inventory system by DC programming based approaches," Computational Optimization and Applications, Springer, vol. 57(2), pages 441-468, March.
    3. Sirmatel, Isik Ilber & Geroliminis, Nikolas, 2018. "Mixed logical dynamical modeling and hybrid model predictive control of public transport operations," Transportation Research Part B: Methodological, Elsevier, vol. 114(C), pages 325-345.
    4. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.
    5. Sönke Behrends & Ruth Hübner & Anita Schöbel, 2018. "Norm bounds and underestimators for unconstrained polynomial integer minimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(1), pages 73-107, February.
    6. Claudia D’Ambrosio & Andrea Lodi, 2013. "Mixed integer nonlinear programming tools: an updated practical overview," Annals of Operations Research, Springer, vol. 204(1), pages 301-320, April.
    7. Ngueveu, Sandra Ulrich, 2019. "Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming-based solution methods," European Journal of Operational Research, Elsevier, vol. 275(3), pages 1058-1071.
    8. Albers, Sönke & Brockhoff, Klaus, 1979. "Optimal product attributes in single choice models," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 67, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    9. Shang, You-lin & Zhang, Lian-sheng, 2008. "Finding discrete global minima with a filled function for integer programming," European Journal of Operational Research, Elsevier, vol. 189(1), pages 31-40, August.
    10. Tiago Andrade & Fabricio Oliveira & Silvio Hamacher & Andrew Eberhard, 2019. "Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming," Journal of Global Optimization, Springer, vol. 73(4), pages 701-722, April.
    11. Leo Liberti & Fabrizio Marinelli, 2014. "Mathematical programming: Turing completeness and applications to software analysis," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 82-104, July.
    12. Jesús A. De Loera & Raymond Hemmecke & Matthias Köppe & Robert Weismantel, 2006. "Integer Polynomial Optimization in Fixed Dimension," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 147-153, February.

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