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On Connections Between Zero-One Integer Programming and Concave Programming Under Linear Constraints

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  • M. Raghavachari

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania and Indian Institute of Management, Ahmedabad, India)

Abstract

Consider the zero-one integer programming problem P 1 i:minimize Z = c ′ x subject to Ax ≦ b , 0 ≦ x i ≦ 1, x j = 0 or 1, j = 1, 2, …, n , where A is an m × n matrix, c ′ = ( c 1 , …, c n ), x ′ = ( x 1 , …, x n ), and b is an m × 1 vector with b ′ = ( b 1 , …, b m ). Assume the elements of A , b , c are all rational. This paper characterizes the feasible solutions of P 1 , shows that P 1 is equivalent to a problem of minimizing a concave quadratic objective function over a convex set, and applies a method developed by Tul to solve such a problem to yield a procedure for the zero-one integer programming problem.

Suggested Citation

  • M. Raghavachari, 1969. "On Connections Between Zero-One Integer Programming and Concave Programming Under Linear Constraints," Operations Research, INFORMS, vol. 17(4), pages 680-684, August.
    • Handle: RePEc:inm:oropre:v:17:y:1969:i:4:p:680-684
    DOI: 10.1287/opre.17.4.680
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    Cited by:

    1. Peiping Shen & Kaimin Wang & Ting Lu, 2020. "Outer space branch and bound algorithm for solving linear multiplicative programming problems," Journal of Global Optimization, Springer, vol. 78(3), pages 453-482, November.
    2. Tao Tan & Yanyan Li & Xingsi Li, 2011. "A Smoothing Method for Zero–One Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 65-77, July.
    3. Jonathan Eckstein & Mikhail Nediak, 2005. "Depth-Optimized Convexity Cuts," Annals of Operations Research, Springer, vol. 139(1), pages 95-129, October.
    4. Wendel Melo & Marcia Fampa & Fernanda Raupp, 2018. "Integrality gap minimization heuristics for binary mixed integer nonlinear programming," Journal of Global Optimization, Springer, vol. 71(3), pages 593-612, July.
    5. Vasilios I. Manousiouthakis & Neil Thomas & Ahmad M. Justanieah, 2011. "On a Finite Branch and Bound Algorithm for the Global Minimization of a Concave Power Law Over a Polytope," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 121-134, October.
    6. Ma, Cheng & Zhang, Liansheng, 2015. "On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 642-656.
    7. Filippo Fabiani & Barbara Franci, 2023. "On Distributionally Robust Generalized Nash Games Defined over the Wasserstein Ball," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 298-309, October.
    8. Hager, William W. & Hungerford, James T., 2015. "Continuous quadratic programming formulations of optimization problems on graphs," European Journal of Operational Research, Elsevier, vol. 240(2), pages 328-337.
    9. M. Santis & F. Rinaldi, 2012. "Continuous Reformulations for Zero–One Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 75-84, April.
    10. H. P. Benson, 2008. "Global Maximization of a Generalized Concave Multiplicative Function," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 105-120, April.
    11. Stefano Lucidi & Francesco Rinaldi, 2010. "An Exact Penalty Global Optimization Approach for Mixed-Integer Programming Problems," DIS Technical Reports 2010-17, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".
    12. Marianna De Santis & Francesco Rinaldi, 2010. "Continuous reformulations for zero-one programming problems," DIS Technical Reports 2010-16, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".

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