IDEAS home Printed from https://ideas.repec.org/a/wly/navres/v43y1996i6p765-795.html
   My bibliography  Save this article

Deterministic algorithms for constrained concave minimization: A unified critical survey

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/(SICI)1520-6750(199609)43:63.0.CO;2-2
    Download Restriction: no
    File URL: https://libkey.io/10.1002/(SICI)1520-6750(199609)43:63.0.CO;2-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
    5. 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.
    6. 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.
    7. 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.
    8. Hamdy A. Taha, 1973. "Concave minimization over a convex polyhedron," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 20(3), pages 533-548, September.
    9. 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.
    10. Richard M. Soland, 1974. "Optimal Facility Location with Concave Costs," Operations Research, INFORMS, vol. 22(2), pages 373-382, April.
    11. 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.
    12. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    13. 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.
    14. Katta G. Murty, 1968. "Solving the Fixed Charge Problem by Ranking the Extreme Points," Operations Research, INFORMS, vol. 16(2), pages 268-279, April.
    15. 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.
    16. 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.
    17. James E. Falk & Karla L. Hoffman, 1986. "Concave Minimization Via Collapsing Polytopes," Operations Research, INFORMS, vol. 34(6), pages 919-929, December.
    18. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Duan Li & Xiaoling Sun & Ken McKinnon, 2005. "An Exact Solution Method for Reliability Optimization in Complex Systems," Annals of Operations Research, Springer, vol. 133(1), pages 129-148, January.
    3. Marcus Porembski, 2004. "Cutting Planes for Low-Rank-Like Concave Minimization Problems," Operations Research, INFORMS, vol. 52(6), pages 942-953, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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.
    4. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2021. "A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems," IIMA Working Papers WP 2021-03-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
    5. 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.
    6. Harold P. Benson, 2004. "Concave envelopes of monomial functions over rectangles," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(4), pages 467-476, June.
    7. M. Vanhoucke, 2002. "Optimal Due Date Assignment In Project Scheduling," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 02/159, Ghent University, Faculty of Economics and Business Administration.
    8. 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.
    9. H. X. Phu & V. M. Pho & P. T. An, 2011. "Maximizing Strictly Convex Quadratic Functions with Bounded Perturbations," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 1-25, April.
    10. Nonas, Sigrid Lise & Thorstenson, Anders, 2000. "A combined cutting-stock and lot-sizing problem," European Journal of Operational Research, Elsevier, vol. 120(2), pages 327-342, January.
    11. Vanhoucke, Mario & Demeulemeester, Erik & Herroelen, Willy, 2003. "Progress payments in project scheduling problems," European Journal of Operational Research, Elsevier, vol. 148(3), pages 604-620, August.
    12. Torki, Abdolhamid & Yajima, Yatsutoshi & Enkawa, Takao, 1996. "A low-rank bilinear programming approach for sub-optimal solution of the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 94(2), pages 384-391, October.
    13. S. Selcuk Erenguc, 1988. "Multiproduct dynamic lot‐sizing model with coordinated replenishments," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(1), pages 1-22, February.
    14. Wooseung Jang & J. George Shanthikumar, 2002. "Stochastic allocation of inspection capacity to competitive processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(1), pages 78-94, February.
    15. Benson, Harold P., 2006. "Fractional programming with convex quadratic forms and functions," European Journal of Operational Research, Elsevier, vol. 173(2), pages 351-369, September.
    16. Takahito Kuno, 2022. "A revision of the rectangular algorithm for a class of DC optimization problems," Journal of Global Optimization, Springer, vol. 83(2), pages 187-200, June.
    17. Phan Thiên Thach & Hoàng Tuy, 1990. "The relief indicator method for constrained global optimization," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(4), pages 473-497, August.
    18. R. Horst & N. V. Thoai, 1999. "DC Programming: Overview," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 1-43, October.
    19. 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.
    20. Bahman Kalantari & Ansuman Bagchi, 1990. "An algorithm for quadratic zero‐one programs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(4), pages 527-538, August.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:navres:v:43:y:1996:i:6:p:765-795. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1520-6750 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.