Computing

, Volume 39, Issue 4, pp 281–291 | Cite as

Global minimization of indefinite quadratic problems

  • P. M. Pardalos
  • J. H. Glick
  • J. B. Rosen
Contributed Papers

Abstract

A branch and bound algorithm is proposed for finding the global optimum of large-scale indefinite quadratic problems over a polytope. The algorithm uses separable programming and techniques from concave optimization to obtain approximate solutions. Results on error bounding are given and preliminary computational results using the Cray 1S supercomputer as reported.

AMS Subject Classifications

65K05 90C30 

Globale Minimisierung indefiniter quadratischer Probleme

Zusammenfassung

Wir schlagen einen neuen „branch and bound” Algorithmus vor, um globale Optima indefiniter quadratischer Probleme über einem Polytop zu berechnen. Unser Algorithmus benutzt separable Programmierung und Techniken der konkaven Optimierung zur näherungsweisen Berechnung von Lösungen. Ferner wird eine Fehleranalyse durchgeführt und es wird über vorläufige experimentelle Ergebnisse (auf einer Cray 1 S) berichtet.

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References

  1. [1]
    Brayton, R. K., Hachtel, G. D., Sangiovanni-Vincentelli, A. L.: A survey of optimization techniques for integrated-circuit design. Proceedings of the IEEE69/10, 1334–1362 (1981).Google Scholar
  2. [2]
    Ciesielski, M. J., Kinnen, E.: An Analytic Method for Compacting Routing Area in Integrated Circuits. Proceedings of the 19th Design Automation Conference, Las Vegas, NV. (1982) pp. 30–37.Google Scholar
  3. [3]
    Cirina, M.: A class of nonlinear programming test problems. Working paper, Dipart. di Informatica, Torino (1985).Google Scholar
  4. [4]
    Crowder, H., Johnson, E. L., Padberg, M. W.: Solving large-scale zero-one linear programming problems. Oper. Res.31/5, 803–834 (1982).Google Scholar
  5. [5]
    Geoffrion, A.: Generalized Bender's decompositions. J. Optimiz. Theory Applic.10, 237–260 (1972).Google Scholar
  6. [6]
    Kalantari, B.: Large scale concave quadratic minimization and extensions. PhD thesis, Computer Sci. Dept., University of Minnesota (1984).Google Scholar
  7. [7]
    Kedem, G., Watanabe, H.: Optimization Techniques for IC Layout and Compaction. Proceedings IEEE Intern. Conf. in Computer Design: VLSI in Computers (1983) pp. 709–713.Google Scholar
  8. [8]
    Kough, P. F.: The indefinite quadratic programming problem. Oper. Res.27/3, 516–533 (1979).Google Scholar
  9. [9]
    Maling, K., Mueller, S. H., Heller, W. R.: On finding most optimal rectangular package plans. Proceedings of the 19th Design Automation Conference, Las Vegas, NV. (1982) pp. 663–670.Google Scholar
  10. [10]
    Mueller, R. K.: A method for solving the indefinite quadratic programming problem. Manag. Science16/5, 333–339 (1979).Google Scholar
  11. [11]
    Meyer, R. R.: Computational aspects of two-segment separable programming. Math Progr.26, 21–32 (1983).Google Scholar
  12. [12]
    Murtagh, B. A., Saunders, M. A.: MINOS 5.0 User's Guide. Tech. Rep. SOL 83-20, Dept. of Oper. Res., Stanford Univ. (1983).Google Scholar
  13. [13]
    Pardalos, P. M.: Integer, and separable programming techniques for large scale global optimization problems. PhD thesis, Computer Sci. Dept, University of Minnesota (1985).Google Scholar
  14. [14]
    Pardalos, P. M., Rosen, J. B.: Methods for global concave minimization: a bibliographic survey. SIAM Review8/3, 367–379 (1986).Google Scholar
  15. [15]
    Pardalos, P. M., Rosen, J. B.: Global optimization approach to the linear complementarity problem. To appear in SIAM J. Sci. Stat. Computing (1987).Google Scholar
  16. [16]
    Rosen, J. B., Pardalos, P. M.: Global minimization of large scale constrained concave quadratic problems by separable programming. Math. Progr.34, 163–174 (1986).Google Scholar
  17. [17]
    Soukup, J.: Circuit layout. Proceedings of the IEEE.69/10, 1281–1304 (1984).Google Scholar
  18. [18]
    Thakur, L. S.: Error analysis for convex separable programs: the piecewise linear approximation and the bounds on the optimal objective value. SIAM J. Appl. Math.34/4., 704–714 (1978).Google Scholar
  19. [19]
    Thoai, N. V., Tuy, H.: Solving the linear complementary problem through concave programming. Zh. Vychisl. Mat. i. Mat. Fiz.23/3, 602–608 (1983).Google Scholar
  20. [20]
    Tuy, H.: Global minimization of the difference of two convex functions. In: Selected Topics in Operations Research and Mathematical Economics. Lecture Notes Econ. Math. Syst.226, 98–118 (1984).Google Scholar
  21. [21]
    Watanabe, H.: IC Layout Generation and Compaction Using Mathematical Optimization. Ph. D. thesis Comp. Sci. Dept. Rochester Univ. (1984).Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • P. M. Pardalos
    • 1
  • J. H. Glick
    • 2
  • J. B. Rosen
    • 2
  1. 1.Department of Computer ScienceThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Computer Science DepartmentUniversity of MinnesotaMinneapolisUSA

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