Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time

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

Abstract

Non-convex knapsack separable quadratic optimization with compact box constraints is an NP-hard problem. We present tight lower and upper bounding procedures that are computationally-efficient as the problem size grows. The lower bound is based on Lagrangian relaxation, and it is computed in linear-time. When the bound is not an exact global solution, a worst-case bound-quality measure is developed. Moreover, the lower bounding (LB) solution is improved to construct a feasible solution, leading to an upper bound (UB) on the given problem. The UB is based on fixing the convex variables and a subset of non-convex and linear variables at the LB solution, and considering the remaining problem, which is always feasible. The UB is computable with linear-time complexity, and it is the global optimum in certain verifiable cases when the duality gap is zero. In our limited computational experiments, the UB has very small relative gap with an exact global optimum solution when there is a nonzero duality gap. Performance of the bounds is demonstrated through a broad range of randomly generated problem instances and comparisons with existing global and non-global solvers. The proposed method on these indefinite problems of extremely large size is an order of magnitude faster than the alternative solvers, including IBM’s commercial software Cplex12.6.

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

Access this article

Log in via an institution

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

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: \(\min \{0.5d_kx^2_k-c_kx_k\;:\; l_k\le x_k \le u_k \}\).

  2. The code was compiled into MATLAB with Mex codes shared at http://www.mathworks.com/matlabcentral/fileexchange/29453-nth-element by Peter Li. The ISO International Standard ISO/IEC 14882 guarantees the linear time median search of nth-element function.

References

  1. Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows, 4th edn. Wiley, New York (2009)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Cominetti, R., Mascarenhas, W., Silva, P.: A Newton’s method for the continuous quadratic knapsack problem. Math. Program. Comput. 6(2), 151–169 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Edirisinghe, C., Jeong, J.: An efficient global algorithm for a class of indefinite separable quadratic programs. Math. Program. 158(1), 143–173 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Floyd, R.W., Rivest, R.L.: Algorithm 489: the algorithm select for finding the i th smallest of n elements. Commun. ACM 18(3), 173 (1975)

    Article  Google Scholar 

  8. Martello, S., Pisinger, D., Toth, P.: Dynamic programming and strong bounds for the 0–1 knapsack problem. Manag. Sci. 45(3), 414–424 (1999)

    Article  MATH  Google Scholar 

  9. Moré, J.J., Vavasis, S.A.: On the solution of concave knapsack problems. Math. Program. 49(1), 397–411 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Nakagawa, Y., James, R.J.W., Rego, C., Edirisinghe, C.: Entropy-based optimization of nonlinear separable discrete decision models. Manag. Sci. 60(3), 695–707 (2014)

    Article  Google Scholar 

  11. Patriksson, M., Strömberg, C.: Algorithms for the continuous nonlinear resource allocation problem—new implementations and numerical studies. Eur. J. Oper. Res. 243(3), 703–722 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pisinger, D.: Where are the hard knapsack problems? Comput. Oper. Res. 32(9), 2271–2284 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Robinson, A.G., Jiang, N., Lerme, C.S.: On the continuous quadratic knapsack-problem. Math. Program. 55(1), 99–108 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We sincerely thank the anonymous referees for their thorough and insightful reviews and suggestions to improve the exposition of the paper.

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. Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time. Math. Program. 164, 193–227 (2017). https://doi.org/10.1007/s10107-016-1082-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-016-1082-7

Keywords

Mathematics Subject Classification

Search

Navigation