Skip to main content
SpringerLink
Log in

A novel approach for solving multi-parametric problems with nonlinear constraints

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Parametric optimization problems appear in many areas of applications even though most of the existing solution methods for such problems are limited to problems with polyhedral constraints. In this article, a global solution strategy is proposed for a general convex multi-parametric problems with nonlinear constraints and bounded regions. The basic idea of the proposed approach is to obtain an approximate parametric solution based on the sensitivity analysis theory in the interior of the nonlinear feasible region, and on finding analytic parametric solutions on the boundaries of the nonlinear constraints. The method employs a barrier function reformulation technique to construct a barrier multi-parametric problem with polyhedral constraints. The proposed method also provides exact solutions to convex multi-parametric problems whose objective function and constraints are polynomials of up to third-degree in the optimization variables and quadratic in the parameters vector.

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
Fig. 7

Similar content being viewed by others

References

  1. Fiacco, A.V.: Sensitivity analysis for nonlinear programming using penalty methods. Math. Program. 10, 287–311 (1976)

    Article  MATH  Google Scholar 

  2. Domínguez, L.F., Narciso, D.A., Pistikopoulos, E.N.: Recent advances in multi-parametric nonlinear programming. Comput. Chem. Eng. 34(5), 707–716 (2010)

    Article  Google Scholar 

  3. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)

    Article  MATH  Google Scholar 

  4. Dua, V., Bozinis, N.A., Pistikopoulos, E.N.: A multi-parametric programming approach for mixed-integer quadratic engineering problems. Comput. Chem. Eng. 26, 715–733 (2002)

    Article  Google Scholar 

  5. Dua, V., Pistikopoulos, E.N.: Algorithms for the solution of multi-parametric mixed-integer nonlinear optimization problems. Ind. Eng. Chem. Res. 38(10), 3976–3987 (1999)

    Article  Google Scholar 

  6. Acevedo, J., Salgueiro, M.: An efficient algorithm for convex multi-parametric nonlinear programming problems. Ind. Eng. Chem. Res. 42(23), 5883–5890 (2003)

    Article  Google Scholar 

  7. Domínguez, L.F., Pistikopoulos, E.N.: A quadratic approximation-based algorithm for the solution of multiparametric mixed-integer nonlinear programming problems. AIChE J. 59(2), 483–495 (2013)

    Article  Google Scholar 

  8. Johansen, T.A.: On multi-parametric nonlinear programming and explicit nonlinear model predictive control. In: Proceedings of the IEEE Conference on Decision and Control, vol. 5. Las Vegas, NV, pp. 12–15 (2002). IEEE

  9. Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40(2), 293–300 (2004)

    Article  MATH  Google Scholar 

  10. Pistikopoulos, E.N., Georgiadis, M.C., Dua, V.: Parametric Programming & Control from Theory to Practice. In: Computer Aided Chemical Engineering, vol. 24, pp. 569–574. Elsevier, Weinheim (2007)

    Google Scholar 

  11. Kassa, A.M., Kassa, S.M.: Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints. Inter. J. Optim. Cont.: Theories & Applications (IJOCTA) 4(2), 89–98 (2014)

    MATH  Google Scholar 

  12. Domínguez, L.F., Pistikopoulos, E.N.: Quadratic approximation algorithm for multiparametric nonlinear programming problems. Technical report, Imperial College London, London (2009)

    Google Scholar 

  13. Dua, V., Pistikopoulos, E.N.: An algorithm for the solution of multi-parametric mixed integer linear programming problems. Ann. Oper. Res. 99, 123–139 (2000)

    Article  MATH  Google Scholar 

  14. Faísca, N.P., Saraiva, M.P., Rustem, B., Pistikopoulos, E.N.: A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. CMS 6(4), 377–397 (2009)

    Article  MATH  Google Scholar 

  15. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic, New York (1983)

    MATH  Google Scholar 

  16. Fiacco, A.V., Kyparisis, J.: Convexity and concavity properties of the optimal value function in parametric nonlinear programming. J. Optim. Theory Appl. 48(1), 95–126 (1986)

    Article  MATH  Google Scholar 

  17. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. J. Wiley, New York (1968)

    MATH  Google Scholar 

  18. Pappas, I., Diangelakis, N.A., Pistikopoulos, E.N.: The exact solution of multiparametric quadratically constrained quadratic programming problems. J. Global Optim. 79(1), 59–85 (2021)

    Article  MATH  Google Scholar 

  19. Charitopoulos, V.M., Dua, V.: Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming. AIChE J. 62(9), 3441–3460 (2016)

    Article  Google Scholar 

  20. Dua, V.: Mixed integer polynomial programming. Computers & Chemical Engineering 72, 387–394 (2015)

    Article  Google Scholar 

  21. Fotiou, I.A., Rostalski, P., Parrilo, P.A., Morari, M.: Parametric optimization and optimal control using algebraic geometry methods. Int. J. Control 79(11), 1340–1358 (2007)

    Article  MATH  Google Scholar 

  22. Pistikopoulos, E.N., Diangelakis, N.A., Oberdieck, R.: Multi-parametric Optimization and Control, 1st edn. John Wiley & Sons Inc, Hoboken, NJ (2021)

    MATH  Google Scholar 

  23. Oberdieck, R., Diangelakis, N.A., Pistikopoulos, E.N.: Explicit model predictive control: A connected-graph approach. Automatica 76, 103–112 (2017)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The research of the first author was supported by the International Science Program (ISP) of Sweden, through a research project at the Department of Mathematics, Addis Ababa University.

Author information

Author notes

  1. S. M. Kassa: This author contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Dire Dawa University, Dire Dawa, Ethiopia

    Addis Belete Zewde

  2. Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, P/Bag 16, Palapye, Botswana

    Semu Mitiku Kassa

Authors
  1. Addis Belete Zewde
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Semu Mitiku Kassa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Addis Belete Zewde.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zewde, A.B., Kassa, S.M. A novel approach for solving multi-parametric problems with nonlinear constraints. J Glob Optim 85, 283–313 (2023). https://doi.org/10.1007/s10898-022-01204-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01204-5

Keywords

Mathematics Subject Classification

Search

Navigation