Advertisement

Exact Optimal Solution to Nonseparable Concave Quadratic Integer Programming Problems

  • Fenlan WangEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 279)

Abstract

Nonseparable quadratic integer programming problems have extensive applications in real world and have received considerable attentions. In this paper, a new exact algorithm is presented for nonseparable concave quadratic integer programming problems. This algorithm is of a branch and bound frame, where the lower bound is obtained by solving a quadratic convex programming problem and the branches are partitioned via a special domain cut technique by which the optimality gap is reduced gradually. The optimal solution to the primal problem can be found in a finite number of iterations. Numerical results are also reported to illustrate the efficiency of our algorithm.

Keywords

Nonseparable concave integer programming problems Linear and contour cut Domain partition Quadratic convex programming 

MSC (2010):

90C10 90C26 90C30 

References

  1. 1.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1993)zbMATHGoogle Scholar
  2. 2.
    Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optimiz. 11, 179–188 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benson, H.P., Erengue, S.S.: An algorithm for concave integer minimization over a polyhedron. Nav. Res. Log. 37, 515–525 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bretthauer, K.M., Shetty, B.: The nonlinear resource allocation problem. Oper. Res. 43, 670–683 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bretthauer, K.M., Shetty, B.: The nonlinear knapsack problem-algorithms and applications. Eur. J. Oper. Res. 138, 459–472 (2002a)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bretthauer, K.M., Shetty, B.: A pegging algorithm for the nonlinear resource allocation problem. Comput. Oper. Res. 29, 505–527 (2002b)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cabot, A.V., Erengue, S.S.: A branch and bound algorithm for solving a class of nonlinear integer programming problems. Nav. Res. Log. 33, 559–567 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guignard, M., Kim, S.: Lagrangian decomposition: a model yielding stronger lagrangian relaxation bounds. Math. Program. 33, 262–273 (1987)Google Scholar
  9. 9.
    Hochbaum, D.: A nonlinear knapsack problem. Oper. Res. Lett. 17, 103–110 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  11. 11.
    Ibaraki, T., Katoh, N.: Resource Allocation Problems: Algorithmic Approaches. MIT Press, Cambridge, Mass (1988)Google Scholar
  12. 12.
    Kodialam, M.S., Luss, H.: Algorithm for separable nonlinear resource allocation problems. Oper. Res. 46, 272–284 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optimiz. 12, 756–769 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Li, D., Sun, X.L., Wang, F.L.: Convergent Lagrangian and contour cut method for nonlinear integer programming with a quadratic objective function. SIAM J. Optimiz. 17, 372–400 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Li, D., Sun, X.L., Wang, J., McKinnon, K.: Convergent Lagrangian and domain cut method for nonlinear knapsack problems. Comput. Optim. Appl. 42, 67–104 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Marsten, R.E., Morin, T.L.: A hybrid approach to discrete mathematical programming. Math. Program. 14, 21–40 (1978)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mathur, K., Salkin, H.M., Morito, S.: A branch and search algorithm for a class of nonlinear knapsack problems. Oper. Res. Lett. 2, 55–60 (1983)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Michelon, P., Maculan, N.: Lagrangian decomposition for integer nonlinear programming with linear constraints. Math. Program. 52, 303–313 (1991)CrossRefGoogle Scholar
  19. 19.
    Michelon, P., Maculan, N.: Lagrangian methods for 0–1 quadratic programming. Discre. Appl. Math. 42, 257–269 (1993)CrossRefGoogle Scholar
  20. 20.
    Pardalos, P.M., Rosen, J.B.: Reduction of nonlinear integer separable programming problems. Int. J. Comput. Math. 24, 55–64 (1988)CrossRefGoogle Scholar
  21. 21.
    Sun, X.L., Li, D.: Optimality condition and branch and bound algorithm for constrained redundancy optimization in series systems. Optim. Eng. 3, 53–65 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sun, X.L., Wang, F.L., Li, D.: Exact algorithm for concave knapsack problems: Linear underestination and partition method. J. Global Optim. 33, 15–30 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, F.L., Sun, X.L.: A Lagrangian decomposition and domain cut algorithm for nonseparable convex knapsack problems. Oper. Res. Trans. 8, 45–53 (2004)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of ScienceNanjing University of Aeronautics and AstronauticsNanjingChina

Personalised recommendations