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A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems (revised as on 12/08/2021)

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  • Sinha, Ankur
  • Das, Arka
  • Anand, Guneshwar
  • Jayaswal, Sachin

Abstract

In this article, we discuss an exact algorithm for mixed integer concave minimization problems. A piece wise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single-level formulation with the help of Karush-Kuhn-Tucker(KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM. Multiple bilevel programs, When solved over iterations, guarantee convergence to the exact optimum of the original problem. Though the algorithm is general and can be applied to any optimization problem with concave function(s), in this paper, we solve two common classes of operations and supply chain problems; namely, the concave knapsack problem, and the concave production transportation problem. The computational experiments indicate that our proposed approach out performs the customized methods that have been used in the literature to solve the two classes of problems by an order of magnitude in most of the test cases.

Suggested Citation

  • Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2021. "A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems (revised as on 12/08/2021)," IIMA Working Papers WP 2021-03-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
    • Handle: RePEc:iim:iimawp:14658
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    References listed on IDEAS

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    1. Harold P. Benson, 1985. "A finite algorithm for concave minimization over a polyhedron," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 32(1), pages 165-177, February.
    2. 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.
    3. Philip B. Zwart, 1974. "Global Maximization of a Convex Function with Linear Inequality Constraints," Operations Research, INFORMS, vol. 22(3), pages 602-609, June.
    4. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    5. Hamdy A. Taha, 1973. "Concave minimization over a convex polyhedron," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 20(3), pages 533-548, September.
    6. Strekalovsky, Alexander S., 2015. "On local search in d.c. optimization problems," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 73-83.
    7. A. Victor Cabot & S. Selcuk Erenguc, 1986. "A branch and bound algorithm for solving a class of nonlinear integer programming problems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(4), pages 559-567, November.
    8. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, April.
    9. Katta G. Murty, 1968. "Solving the Fixed Charge Problem by Ranking the Extreme Points," Operations Research, INFORMS, vol. 16(2), pages 268-279, April.
    10. Bretthauer, Kurt M. & Ross, Anthony & Shetty, Bala, 1999. "Nonlinear integer programming for optimal allocation in stratified sampling," European Journal of Operational Research, Elsevier, vol. 116(3), pages 667-680, August.
    11. Michelon, Philippe & Veilleux, Louis, 1996. "Lagrangean methods for the 0-1 Quadratic Knapsack Problem," European Journal of Operational Research, Elsevier, vol. 92(2), pages 326-341, July.
    12. Kurt M. Bretthauer & Bala Shetty, 1995. "The Nonlinear Resource Allocation Problem," Operations Research, INFORMS, vol. 43(4), pages 670-683, August.
    13. 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.
    14. Xiang Li & Asgeir Tomasgard & Paul I. Barton, 2011. "Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 425-454, December.
    15. Sharp, J Frank & Snyder, James C & Greene, James H, 1970. "A Decomposition Algorithm for Solving the Multifacility Production-Transportation Problem with Nonlinear Production Costs," Econometrica, Econometric Society, vol. 38(3), pages 490-506, May.
    16. Marshall L. Fisher, 2004. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 50(12_supple), pages 1861-1871, December.
    17. Richard M. Soland, 1971. "An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints," Management Science, INFORMS, vol. 17(11), pages 759-773, July.
    18. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
    19. Peter J. Kolesar, 1967. "A Branch and Bound Algorithm for the Knapsack Problem," Management Science, INFORMS, vol. 13(9), pages 723-735, May.
    20. Gallo, Giorgio & Sandi, Claudio & Sodini, Claudio, 1980. "An algorithm for the min concave cost flow problem," European Journal of Operational Research, Elsevier, vol. 4(4), pages 248-255, April.
    21. Richard M. Soland, 1974. "Optimal Facility Location with Concave Costs," Operations Research, INFORMS, vol. 22(2), pages 373-382, April.
    22. Dijkhuizen, G. & Faigle, U., 1993. "A cutting-plane approach to the edge-weighted maximal clique problem," European Journal of Operational Research, Elsevier, vol. 69(1), pages 121-130, August.
    23. Gabriel R. Bitran & Devanath Tirupati, 1989. "Tradeoff Curves, Targeting and Balancing in Manufacturing Queueing Networks," Operations Research, INFORMS, vol. 37(4), pages 547-564, August.
    24. Marshall L. Fisher, 2004. "Comments on ÜThe Lagrangian Relaxation Method for Solving Integer Programming ProblemsÝ," Management Science, INFORMS, vol. 50(12_supple), pages 1872-1874, December.
    25. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
    26. Park, Kyungchul & Lee, Kyungsik & Park, Sungsoo, 1996. "An extended formulation approach to the edge-weighted maximal clique problem," European Journal of Operational Research, Elsevier, vol. 95(3), pages 671-682, December.
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