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Deterministic algorithms for constrained concave minimization: A unified critical survey

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  • Harold P. Benson

Abstract

The purpose of this article is to present a unified critical survey of the deterministic solution algorithms for constrained concave minimization. To unify and streamline the survey, the article first describes four fundamental techniques that serve as building blocks for most of these algorithms. These four techniques are extreme point ranking, concavity cut reduction, outer approximation, and branch and bound. Using these descriptions, the article then surveys the deterministic algorithms for constrained concave minimization by grouping them into three categories of approaches. These categories are enumeration, successive approximation, and successive partitioning. This strategy provides a means of efficiently conveying the essential mechanics and the relative strengths and weaknesses of most of the well‐known concave minimization algorithms. © 1996 John Wiley & Sons, Inc.

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  • Harold P. Benson, 1996. "Deterministic algorithms for constrained concave minimization: A unified critical survey," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(6), pages 765-795, September.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:6:p:765-795
    DOI: 10.1002/(SICI)1520-6750(199609)43:63.0.CO;2-2
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    1. Timothy L. Shaftel & Gerald L. Thompson, 1977. "A Simplex-Like Algorithm for the Continuous Modular Design Problem," Operations Research, INFORMS, vol. 25(5), pages 788-807, October.
    2. 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.
    3. Katta G. Murty, 1968. "Solving the Fixed Charge Problem by Ranking the Extreme Points," Operations Research, INFORMS, vol. 16(2), pages 268-279, April.
    4. Lakshman S. Thakur, 1991. "Domain Contraction in Nonlinear Programming: Minimizing a Quadratic Concave Objective Over a Polyhedron," Mathematics of Operations Research, INFORMS, vol. 16(2), pages 390-407, May.
    5. 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.
    6. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
    7. B. Kalantari & J. B. Rosen, 1987. "An Algorithm for Global Minimization of Linearly Constrained Concave Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 544-561, August.
    8. J. B. Rosen, 1983. "Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 215-230, May.
    9. M. Hamami & S. E. Jacobsen, 1988. "Exhaustive Nondegenerate Conical Processes for Concave Minimization on Convex Polytopes," Mathematics of Operations Research, INFORMS, vol. 13(3), pages 479-487, August.
    10. A. Victor Cabot & Richard L. Francis, 1970. "Solving Certain Nonconvex Quadratic Minimization Problems by Ranking the Extreme Points," Operations Research, INFORMS, vol. 18(1), pages 82-86, February.
    11. James E. Falk & Karla L. Hoffman, 1986. "Concave Minimization Via Collapsing Polytopes," Operations Research, INFORMS, vol. 34(6), pages 919-929, December.
    12. Nguyen Van Thoai & Hoang Tuy, 1980. "Convergent Algorithms for Minimizing a Concave Function," Mathematics of Operations Research, INFORMS, vol. 5(4), pages 556-566, November.
    13. Hamdy A. Taha, 1973. "Concave minimization over a convex polyhedron," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 20(3), pages 533-548, September.
    14. Reiner Horst, 1990. "Deterministic methods in constrained global optimization: Some recent advances and new fields of application," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(4), pages 433-471, August.
    15. Richard M. Soland, 1974. "Optimal Facility Location with Concave Costs," Operations Research, INFORMS, vol. 22(2), pages 373-382, April.
    16. 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.
    17. Patrick McKeown, 1975. "Technical Note—A Vertex Ranking Procedure for Solving the Linear Fixed-Charge Problem," Operations Research, INFORMS, vol. 23(6), pages 1183-1191, December.
    18. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    19. 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.
    20. 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.
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    Cited by:

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    3. 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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