Mathematical Programming

, Volume 34, Issue 2, pp 163–174 | Cite as

Global minimization of large-scale constrained concave quadratic problems by separable programming

  • J. B. Rosen
  • P. M. Pardalos


The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesxR n , while the linear part is in terms ofyRk. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be ≤ 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteedε-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.

Key words

Global Minimization Separable Programming Quadratic Programming Large-Scale 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M.S. Bazaraa and H.D. Sherali, “On the use of exact and heuristic cutting plane methods for the quadratic assignment problem“,Journal of Operational Research Society 33 (1982) 991–1003.Google Scholar
  2. [2]
    H. Crowder, E.L. Johnson and M.W. Padberg, “Solving large-scale zero–one linear programming problems“,Operations Research 31 (1982) 803–834.Google Scholar
  3. [3]
    J.E. Falk and K.R. Hoffman, “A successive underestimating method for concave minimization problems“,Mathematics of Operations Research 1 (1976) 251–259.Google Scholar
  4. [4]
    A.M. Frieze, “A bilinear programming formulation of the 3-dimensional assignment problem“,Mathematical Programming 7 (1974) 376–379.Google Scholar
  5. [5]
    C.D. Heising-Goodman, “A survey of methodology for the global minimization of concave functions subject to convex constraints“,OMEGA, International Journal of Management Science 9 (1981) 313–319.Google Scholar
  6. [6]
    R. Horst, “An algorithm for nonconvex programming problems“,Mathematical Programming 10 (1976) 312–321.Google Scholar
  7. [7]
    B. Kalantari, “Large scale concave quadratic minimization and extensions”, Ph.D. thesis, Computer Science Department, University of Minnesota (Minneapolis, MN, 1984).Google Scholar
  8. [8]
    B. Kalantari and J.B. Rosen, “Construction of large scale global minimum concave quadratic test problems”, to be published inJournal of Optimization Theory and Applications (1986).Google Scholar
  9. [9]
    H. Konno, “Maximizing a convex quadratic function subject to linear constraints“,Mathematical Programming 11 (1976) 117–127.Google Scholar
  10. [10]
    E.L. Lawler, “The quadratic assignment problem“,Management Science 9 (1963) 586–599.Google Scholar
  11. [11]
    O.L. Mangasarian, “Characterization of linear complementarity problems as linear programs“,Mathematical Programming Study 7 (1978) 74–87.Google Scholar
  12. [12]
    P.M. Pardalos and J.B. Rosen, “Methods for global concave minimization: A bibliographic survey”, to appear inSIAM Review (1986).Google Scholar
  13. [13]
    M. Raghavachari, “On connections between zero–one integer programming and concave programming under linear constraints“,Operations Research 17 (1969) 680–684.Google Scholar
  14. [14]
    J.B. Rosen, “Global minimization of a linearly constrained concave function by partition of feasible domain“,Mathematics of Operations Research 8 (1983) 215–230.Google Scholar
  15. [15]
    J.B. Rosen, “Performance of approximate algorithms for global minimization“,Mathematical Programming Study 22 (1984) 231–236.Google Scholar
  16. [16]
    J.B. Rosen, “Computational solution of large scale constrained global minimization problems“, in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization 1984 (SIAM, Philadelphia, PA, 1985) pp. 263–271.Google Scholar
  17. [17]
    L. Schrage,Linear integer and quadratic programming with LINDO (Scientific Press, Palo Alto, CA, 1984).Google Scholar
  18. [18]
    B.T. Smith, J. Boyle, B. Garbow, Y. Ikebe, V. Klema and C. Moler,Matrix eigensystem routines-EISPACK guide, Lecture notes in computer science, Vol. 6 (Springer-Verlag, New York, 1976).Google Scholar
  19. [19]
    N.V. Thoai and H. Tuy, “Convergent algorithms for minimizing a concave function“,Mathematics of Operations Research 5 (1980) 556–566.Google Scholar
  20. [20]
    H. Tuy, “Concave programming under linear constraints“,Doklady Akademii Nauk SSSR 159, 32–35 (English translation inSoviet Mathematics Doklady 5 (1964) 1437–1440).Google Scholar
  21. [21]
    H. Tuy, “Global minimization of a difference of two convex functions“, in:Selected topics in operations research and mathematical economics. Lecture Notes in Economics and Mathematical Systems 226 (1984) 98–118.Google Scholar
  22. [22]
    H. Watanabe, “IC layout generation and compaction using mathematical optimization”, Ph.D. Thesis, Computer Science Department, University of Rochester (Rochester, NY, 1984).Google Scholar
  23. [23]
    N. Zilverberg, “Global minimization for large scale linearly constrained systems”, Ph.D. Thesis, Computer Science Department, University of Minnesota (Minneapolis, MN, 1983).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

  • J. B. Rosen
    • 1
  • P. M. Pardalos
    • 1
  1. 1.Computer Science DepartmentUniversity of MinnesotaMinneapolisUSA

Personalised recommendations