Skip to main content

Local minima for indefinite quadratic knapsack problems

  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the complexity of finding a local minimum for the nonconvex Quadratic Knapsack Problem. Global minimization for this example of quadratic programming is NP-hard. Moré and Vavasis have investigated the complexity of local minimization for the strictly concave case of QKP; here we extend their algorithm to the general indefinite case. Our main result is an algorithm that computes a local minimum in O(n(logn)2) steps. Our approach involves eliminating all but one of the convex variables through parametrization, yielding a nondifferentiable problem. We use a technique from computational geometry to address the nondifferentiable problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • G.R. Bitran and A.C. Hax, “Disaggregation and resource allocation using convex knapsack problems with bounded variables,”Management Science 27 (1981) 431–441.

    Google Scholar 

  • P. Brucker, “An O(n) algorithm for quadratic knapsack problems,”Operations Research Letters 3 (1984) 163–166.

    Google Scholar 

  • P.H. Calamai and J.J. Moré, “Quasi-Newton updates with bounds,”SIAM Journal on Numerical Analysis 24 (1987) 1434–1441.

    Google Scholar 

  • B. Chazelle and L.J. Guibas, “Visibility and intersection problems in plane geometry,”ACM Symposium on Computational Geometry, 1985, pp. 135–146.

  • R.W. Cottle, S.G. Duval and K. Zikan, “A Lagrangean relaxation algorithm for the constrained matrix problem,”Naval Research Logistics Quarterly 33 (1986) 55–76.

    Google Scholar 

  • R. Fletcher,Practical Methods of Optimization, Vol. 2 (Wiley, Chichester, 1981).

    Google Scholar 

  • P.E. Gill, W. Murray and M.H Wright,Practical Optimization (Academic Press, London, 1981).

    Google Scholar 

  • R. Helgason, J. Kennington and H. Lall, “A polynomially bounded algorithm for a singly constrained quadratic program,”Mathematical Programming 18 (1980) 338–343.

    Google Scholar 

  • D.S. Johnson, C.H. Papadimitriou and M. Yannakakis, “How easy is local search?”Journal of Computer and System Sciences 37 (1988) 79–100.

    Google Scholar 

  • M.W. Krentel, “On finding locally optimal solutions,” to appear inSIAM Journal on Computing.

  • R. Kurata, “Notes on parametric quadratic programming,”Journal of the Operations Research Society of Japan 8 (1966) 150–153.

    Google Scholar 

  • H.M. Markowitz, “The optimization of a quadratic function subject to linear constraints,”Naval Research Logistics. Quarterly 3 (1956) 111–133.

    Google Scholar 

  • J.J. Moré and S.A. Vavasis, “On the solution of concave knapsack problems,”Mathematical Programming 49 (1991) 397–411.

    Google Scholar 

  • K.G. Murty and S.N. Kabadi, “Some NP-complete problems in quadratic and nonlinear programming,”Mathematical Programming 39 (1987) 117–129.

    Google Scholar 

  • P.M. Pardalos and N. Kovoor, “An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds,”Mathematical Programming 46 (1990) 321–328.

    Google Scholar 

  • P.M. Pardalos, Y. Ye and C.-G. Han, “Algorithms for the solution of quadratic knapsack problems,” Technical Report CS-89-10 Department of Computer Science Pennsylvania State University (University Park, PA, 1989).

    Google Scholar 

  • P.M. Pardalos and G.P. Rodgers, “A branch and bound algorithm for the maximum clique problem,”Mathematical Programming, to appear.

  • C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization (Prentice Hall, Englewood Cliffs, NJ, 1982).

    Google Scholar 

  • S. Sahni, “Computationally related problems,”SIAM Journal on Computing 3 (1974) 262–279.

    Google Scholar 

  • A.A. Schäffer and M. Yannakakis, “Simple local search problems that are hard to solve,” unpublished manuscript, Rice University (Houston, TX, 1989).

    Google Scholar 

  • J. Sun, “Tracing the characteristic curve of a quadratic black box,” unpublished manuscript. Also, “On a quadratic program with a single parameter,” Third SIAM Optimization Conference, Boston MA 1989, poster presentation.

  • C. van de Panne,Methods for Linear and Quadratic Programming (North-Holland, Amsterdam, 1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, Department of Energy, under contract W-31-109-Eng-38, in part by a Fannie and John Hertz Foundation graduate fellowship, and in part by Department of Energy grant DE-FG02-86ER25013.A000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavasis, S.A. Local minima for indefinite quadratic knapsack problems. Mathematical Programming 54, 127–153 (1992). https://doi.org/10.1007/BF01586048

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01586048

Key words