Skip to main content
SpringerLink
Log in

An algorithm to solve multi-objective integer quadratic programming problem

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The multi-objective integer programming problem often occurs in multi-criteria decision-making situations, where the decision variables are integers. In the present paper, we have discussed an algorithm for finding all efficient solutions of a multi-objective integer quadratic programming problem. The proposed algorithm is based on the aspect that efficient solutions of a multi-objective integer quadratic programming problem can be obtained by enumerating ranked solutions of an integer quadratic programming problem. For determining ranked solutions of an integer quadratic programming problem, we have constructed a related integer linear programming problem and from ranked solutions of this integer linear programming problem, ranked solutions of the original integer quadratic programming problem are generated. Theoretically, we have shown that the developed method generates the set of all efficient solutions in a finite number of steps, and numerically we have elaborated the working of our algorithm and compared our results with existing algorithms. Further, we have analyzed that the developed method is efficient for solving a multi-objective integer quadratic programming problem with a large number of constraints, variables and objectives.

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

Similar content being viewed by others

References

  • Arora, R., & Arora, S. R. (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem. Opsearch, 52(2), 367–81.

    Article  MathSciNet  Google Scholar 

  • Ansary, M.A.T., & Panda, G. (2020). A sequential quadratic programming method for constrained multi-objective optimization problems. Journal of Applied Mathematics and Computing, 64(1), 379–397.

  • Batista, A. C., Batista, L. S., & Adriano, R. (2021). A quadratic programming approach for microwave imaging. IEEE Transactions on Antennas and Propagation. 69(8), 4923–4934. https://doi.org/10.1109/TAP.2021.3060092

  • Buchheim, C., Caprara, A., & Lodi, A. (2012). An effective branch-and-bound algorithm for convex quadratic integer programming. Mathematical Programming, 135(1), 369–395.

    Article  MathSciNet  Google Scholar 

  • Cambini, R., & Sodini, C. (2002). A finite algorithm for a particular D.C. quadratic programming problem. Annals of Operations Research, 117(1), 33–49.

    Article  MathSciNet  Google Scholar 

  • Capinski, M., & Zastawniak, T. (2003). Mathematics for Finance. An Introduction to Financial Engineering, Springer London.

  • Day, R. O., Kleeman, M. P., & Lamont, G. B. (2003). Solving the multi-objective quadratic assignment problem using a fast messy genetic algorithm. In Proceedings of the 2003 Congress on Evolutionary Computation, (vol. 3, pp. 2277-22833).

  • Drici, W., & Moulai, M. (2018). An exact method for solving multi-objective integer indefinite quadratic programs. Optimization Methods and Software. https://doi.org/10.1080/10556788.2018.1560443

    Article  Google Scholar 

  • Erenguc, S. S., & Benson, H. P. (1991). An algorithm for indefinite integer quadratic programming. Computers & Mathematics with Applications, 21(6–7), 99–106.

    Article  MathSciNet  Google Scholar 

  • Fiaschi, L., & Cococcioni, M. (2021). A non-Archimedean interior point method for solving lexicographic multi-objective quadratic programming problems. arXiv preprint arXiv:2110.15658.

  • Geissel, S., Graf, H., & Herbinger, Seifried F. T. (2022). Portfolio optimization with optimal expected utility risk measures. Annals of Operations Research, 309, 59–77. https://doi.org/10.1007/s10479-021-04403-7

    Article  MathSciNet  Google Scholar 

  • Ghaffari-Hadigheh, A., Romanko, O., & Terlaky, T. (2010). Bi-parametric convex quadratic optimization. Optimization Methods & Software, 25(2), 229–245.

    Article  MathSciNet  Google Scholar 

  • Grudinin, N. (1998). Reactive power optimization using successive quadratic programming method. IEEE Transactions on Power Systems, 13, 1219–1225.

    Article  ADS  Google Scholar 

  • Gupta, O. K. (1995). Applications of quadratic programming. Journal of Information and Optimization Sciences, 16, 177–194.

    Article  MathSciNet  Google Scholar 

  • Gupta, R., Bandopadhyaya, L., & Puri, M. C. (1996). Ranking in quadratic integer programming problems. European Journal of Operational Research, 95(1), 231–236.

    Article  Google Scholar 

  • Jackson, M., & Staunton, M. D. (1999). Quadratic programming applications in finance using excel. Journal of the Operational Research Society, 50, 1256–1266.

  • Jain, E., Dahiya, K., & Verma, V. (2018). Integer quadratic fractional programming problems with bounded variables. Annals of Operations Research, 269(1), 269–295.

    Article  MathSciNet  Google Scholar 

  • Knowles, J. D., & Corne, D. (2002). Towards landscape analyses to inform the design of hybrid local search for the multiobjective quadratic assignment problem. HIS, 87, 271–279.

    Google Scholar 

  • Kushwah, P., & Sharma, V. (2020). A note on solving multi-objective integer indefinite quadratic fractional programs. Annals of Operations Research, 289, 459–462.

    Article  MathSciNet  Google Scholar 

  • Lachhwani, K. (2012). Fuzzy goal programming approach to multi objective quadratic programming problem. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 82(4), 317–322.

    Article  MathSciNet  Google Scholar 

  • Martos, B. (1965). The direct power of adjacent vertex programming methods. Management Science, 12(3), 241–252.

    Article  MathSciNet  Google Scholar 

  • Mekhilef, A., Moulai, M., & Drici, W. (2021). Solving multi-objective integer indefinite quadratic fractional programs. Annals of Operations Research, 296(1), 821–840.

    Article  MathSciNet  Google Scholar 

  • Oberdieck, R., & Pistikopoulos, E. N. (2016). Multi-objective optimization with convex quadratic cost functions: A multi-parametric programming approach. Computers & Chemical Engineering, 85, 36–39.

    Article  CAS  Google Scholar 

  • Ouaïl, F. Z., & Chergui, MEl.-A. (2018). A branch-and-cut technique to solve multiobjective integer quadratic programming problems. Annals of Operations Research, 267(1–2), 431–446.

    Article  MathSciNet  Google Scholar 

  • Pappas, I., Diangelakis, N. A., & Pistikopoulos, E. N. (2021). The exact solution of multiparametric quadratically constrained quadratic programming problems. Journal of Global Optimization, 79(1), 59–85.

    Article  MathSciNet  Google Scholar 

  • Pramanik, S., & Dey, P. P. (2011). Multi-objective quadratic programming problem: A priority based fuzzy goal programming. International Journal of Computer Applications, 26(10), 30–35.

    Article  Google Scholar 

  • Qi, Y. (2017). On the criterion vectors of lines of portfolio selection with multiple quadratic and multiple linear objectives. Central European Journal of Operations Research, 25(1), 145–158.

    Article  MathSciNet  Google Scholar 

  • Sharma, V., Dhaiya, K., & Verma, V. (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems. Optimization, 66, 1913–1929.

    Article  MathSciNet  Google Scholar 

  • Shim, J. K. (1983). A survey of quadratic programming applications to business and economics. International Journal of Systems Science, 4, 105–115.

    Article  MathSciNet  Google Scholar 

  • Steuer, R. E., Wimmer, M., & Hirschberger, M. (2013). Overviewing the transition of Markowitz bi-criterion portfolio selection to tri-criterion portfolio selection. Journal of Business Economics, 83(1), 61–85.

    Article  Google Scholar 

  • Sylva, J., & Crema, A. (2008). Enumerating the set of non-dominated vectors in multiple objective integer linear programming. RAIRO-Operations Research, 42(3), 371–387.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the computational LAB and library facilities provided under DST-FIST Grant SR/FST/MS-1/2017/13.

Author information

Authors and Affiliations

  1. School of Mathematics, TIET, Patiala, India

    Prerna Kushwah & Vikas Sharma

Authors
  1. Prerna Kushwah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vikas Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vikas Sharma.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kushwah, P., Sharma, V. An algorithm to solve multi-objective integer quadratic programming problem. Ann Oper Res 332, 433–459 (2024). https://doi.org/10.1007/s10479-022-05123-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-022-05123-2

Keywords

Mathematics Subject Classification

Search

Navigation