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An algorithm and new penalties for concave integer minimization over a polyhedron

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  • Kurt M. Bretthauer
  • A. Victor Cabot
  • M. A. Venkataramanan

Abstract

We present a branch‐and‐bound algorithm for globally minimizing a concave function over linear constraints and integer variables. Concave cost functions and integer variables arise in many applications, such as production planning, engineering design, and capacity expansion. To reduce the number of subproblems solved during the branch‐and‐bound search, we also develop a framework for computing new and existing penalties. Computational testing indicates that penalties based on the Tuy cutting plane provide large decreases in solution time for some problems. A combination of Driebeek‐Tomlin and Tuy penalties can provide further decreases in solution time. © 1994 John Wiley & Sons, Inc.

Suggested Citation

  • Kurt M. Bretthauer & A. Victor Cabot & M. A. Venkataramanan, 1994. "An algorithm and new penalties for concave integer minimization over a polyhedron," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(3), pages 435-454, April.
    • Handle: RePEc:wly:navres:v:41:y:1994:i:3:p:435-454
    DOI: 10.1002/1520-6750(199404)41:33.0.CO;2-6
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    References listed on IDEAS

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    1. James E. Falk & Karla R. Hoffman, 1976. "A Successive Underestimation Method for Concave Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 251-259, August.
    2. S. Selcuk Erenguc & Harold P. Benson, 1986. "The interactive fixed charge linear programming problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(2), pages 157-177, May.
    3. Norman J. Driebeek, 1966. "An Algorithm for the Solution of Mixed Integer Programming Problems," Management Science, INFORMS, vol. 12(7), pages 576-587, March.
    4. A. Victor Cabot, 1974. "Variations on a cutting plane method for solving concave minimization problems with linear constraints," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 21(2), pages 265-274, June.
    5. Cohen, Morris A. & Moon, Sangwon, 1991. "An integrated plant loading model with economies of scale and scope," European Journal of Operational Research, Elsevier, vol. 50(3), pages 266-279, February.
    6. Harold P. Benson & S. Selcuk Erenguc, 1990. "An algorithm for concave integer minimization over a polyhedron," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(4), pages 515-525, August.
    7. H. Tuy & T. V. Thieu & Ng. Q. Thai, 1985. "A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set," Mathematics of Operations Research, INFORMS, vol. 10(3), pages 498-514, August.
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    Cited by:

    1. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2023. "A general purpose exact solution method for mixed integer concave minimization problems," European Journal of Operational Research, Elsevier, vol. 309(3), pages 977-992.
    2. Kurt M. Bretthauer, 1994. "A penalty for concave minimization derived from the tuy cutting plane," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(3), pages 455-463, April.
    3. Marcus Porembski, 2008. "On the hierarchy of γ‐valid cuts in global optimization," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(1), pages 1-15, February.
    4. Marcus Porembski, 2004. "Cutting Planes for Low-Rank-Like Concave Minimization Problems," Operations Research, INFORMS, vol. 52(6), pages 942-953, December.

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