Skip to main content
SpringerLink
Log in

An efficient global algorithm for a class of indefinite separable quadratic programs

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We present a global algorithm for indefinite knapsack separable quadratic programs with bound constraints. The upper bounds on variables with nonconvex terms are assumed to be infinite in the algorithmic development. By characterizing optimal solutions of the problem, we enumerate a subset of KKT points to determine a global optimum. The enumeration is made efficient by developing a theory for shrinking and partitioning the search domain of KKT multipliers. The global algorithmic procedure is developed based on interval and point testing techniques that have roots in solving strictly convex problems. Our algorithm has quadratic (worst-case) complexity and its performance is demonstrated on a broad range of randomly generated problem instances and comparisons with existing global and local solvers. It turns out that our method can solve these nonconvex problems of extremely large size in a fraction of the time (relative to commercial software such as CPLEX 12.6) on a desktop computer. Furthermore, a feasible upper bounding scheme is developed for the case when the variables with nonconvex terms are allowed to have finite upper limits, using an iterative application of the above global algorithm, and the computational experience is presented.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. If some \(a_k<0\), then perform the transformation \(x_k\leftarrow -x_k,\,a_k\leftarrow -a_k,\,c_k\leftarrow -c_k,\,l_k\leftarrow -u_k\), and \(u_k\leftarrow -l_k\). If \(a_k=0\), then solve (QP) without \(x_k\) and the optimal value of \(x_k\) is determined by the univariate minimization: \(\min \{0.5d_kx^2_k-c_kx_k\;:\; l_k\le x_k \le u_k \}\).

References

  1. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming, Theory and Applications, 3rd edn. Wiley, New York (2006)

    Book  MATH  Google Scholar 

  2. Bretthauer, K.M., Shetty, B.: The nonlinear knapsack problem-algorithms and applications. Eur. J. Oper. Res. 138(3), 459–472 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Programm. Comput. 4(1), 33–52 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dai, Y.H., Fletcher, R.: New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds. Math. Programm. 106(3), 403–421 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Du, D.Z., Pardalos, P.M.: Handbook of Combinatorial Optimization, vol. 3. Springer, New York (1998)

  6. Helgason, R., Kennington, J.L., Lall, H.: A polynomially bounded algorithm for a singly constrained quadratic program. Math. Programm. 18(3), 338–343 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Horst, R., Pardalos, P.M., Van Thoai, N.: Introduction to Global Optimization. Kluwer Academic Pub, Dordrecht (2000)

    Book  MATH  Google Scholar 

  8. Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, New York (2004)

    Book  MATH  Google Scholar 

  9. Kiwiel, K.: Breakpoint searching algorithms for the continuous quadratic knapsack problem. Math. Programm. 112, 473–491 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Moré, J.J., Vavasis, S.A.: On the solution of concave knapsack problems. Mathe. programm. 49(1), 397–411 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Pardalos, P.M., Glick, J.H., Rosen, J.B.: Global minimization of indefinite quadratic problems. Computing 39(4), 281–291 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pardalos, P.M., Kovoor, N.: An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds. Math. Programm. 46(3), 321–328 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1, 15–23 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sahni, S.: Computationally related problems. SIAM J. Comput. 3(4), 262–279 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vavasis, S.A.: Approximation algorithms for indefinite quadratic programming. Math. Programm. 57(2), 279–311 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Vavasis, S.A.: Local minima for indefinite quadratic knapsack-problems. Math. Programm. 54(2), 127–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yajima, Y.: Quadratic knapsack. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 3159–3161. Springer, New York (2009)

    Chapter  Google Scholar 

Download references

Acknowledgments

Without implicating them, we sincerely thank Stephen Vavasis and Samuel Burer for sharing their computer codes, and Yu-Hong Dai and Roger Fletcher for clarifying certain aspects of their papers. We are also indebted to the anonymous referees for their careful and insightful reviews. All remaining errors in this paper are ours.

Author information

Authors and Affiliations

  1. Rensselaer Polytechnic Institute (RPI), Troy, NY, USA

    Chanaka Edirisinghe

  2. Radford University, Radford, VA, USA

    Jaehwan Jeong

Authors
  1. Chanaka Edirisinghe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaehwan Jeong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chanaka Edirisinghe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Edirisinghe, C., Jeong, J. An efficient global algorithm for a class of indefinite separable quadratic programs. Math. Program. 158, 143–173 (2016). https://doi.org/10.1007/s10107-015-0918-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0918-x

Keywords

Mathematics Subject Classification

Search

Navigation