Skip to main content
SpringerLink
Log in

A comparison of solution approaches for the numerical treatment of or-constrained optimization problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Mathematical programs with or-constraints form a new class of disjunctive optimization problems with inherent practical relevance. In this paper, we provide a comparison of three different solution methods for the numerical treatment of this problem class which are inspired by classical approaches from disjunctive programming. First, we study the replacement of the or-constraints as nonlinear inequality constraints using suitable NCP-functions. Second, we transfer the or-constrained program into a mathematical program with switching or complementarity constraints which can be treated with the aid of well-known relaxation methods. Third, a direct Scholtes-type relaxation of the or-constraints is investigated. A numerical comparison of all these approaches which is based on three essentially different model programs from or-constrained optimization closes the paper.

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. Balas, E.: Disjunctive Programming. Springer, Cham (2018)

    Book  Google Scholar 

  2. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1993)

    MATH  Google Scholar 

  3. Benko, M., Gfrerer, H.: New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints. Optimization 67(1), 1–23 (2018). https://doi.org/10.1080/02331934.2017.1387547

    Article  MathSciNet  MATH  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  5. Clason, C., Deng, Y., Mehlitz, P., Prüfert, U.: Optimal control problems with control complementarity constraints. Optim. Methods Softw. 35(1), 142–170 (2020). https://doi.org/10.1080/10556788.2019.1604705

    Article  MATH  Google Scholar 

  6. Clason, C., Rund, A., Kunisch, K.: Nonconvex penalization of switching control of partial differential equations. Syst. Control Lett. 106, 1–8 (2017). https://doi.org/10.1016/j.sysconle.2017.05.006

    Article  MathSciNet  MATH  Google Scholar 

  7. De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75(3), 407–439 (1996). https://doi.org/10.1007/BF02592192

    Article  MathSciNet  MATH  Google Scholar 

  8. De Luca, T., Facchinei, F., Kanzow, C.: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems. Comput. Optim. Appl. 16(2), 173–205 (2000). https://doi.org/10.1023/A:1008705425484

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002). https://doi.org/10.1007/s101070100263

    Article  MathSciNet  MATH  Google Scholar 

  10. Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85(1), 107–134 (1999). https://doi.org/10.1007/s10107990015a

    Article  MathSciNet  MATH  Google Scholar 

  11. Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7(1), 225–247 (1997). https://doi.org/10.1137/S1052623494279110

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischer, A.: A special Newton-type optimization method. Optimization 24(3–4), 269–284 (1992). https://doi.org/10.1080/02331939208843795

    Article  MathSciNet  MATH  Google Scholar 

  13. Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124(3), 595–614 (2005). https://doi.org/10.1007/s10957-004-1176-x

    Article  MathSciNet  MATH  Google Scholar 

  14. Flegel, M.L., Kanzow, C., Outrata, J.V.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15(2), 139–162 (2007). https://doi.org/10.1007/s11228-006-0033-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17(1), 259–286 (2006). https://doi.org/10.1137/S1052623402407382

    Article  MathSciNet  MATH  Google Scholar 

  16. Fukushima, M., Luo, Z.Q., Pang, J.S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl. 10(1), 5–34 (1998). https://doi.org/10.1023/A:1018359900133

    Article  MathSciNet  MATH  Google Scholar 

  17. Galántai, A.: Properties and construction of NCP functions. Comput. Optim. Appl. 52(3), 805–824 (2012). https://doi.org/10.1007/s10589-011-9428-9

    Article  MathSciNet  MATH  Google Scholar 

  18. Gfrerer, H.: Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints. SIAM J. Optim. 24(2), 898–931 (2014). https://doi.org/10.1137/130914449

    Article  MathSciNet  MATH  Google Scholar 

  19. Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3(3), 227–252 (2002). https://doi.org/10.1023/A:1021039126272

    Article  MathSciNet  MATH  Google Scholar 

  20. Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: nonlinear convex Hull relaxation. Comput. Optim. Appl. 26(1), 83–100 (2003). https://doi.org/10.1023/A:1025154322278

    Article  MathSciNet  MATH  Google Scholar 

  21. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1), 257–288 (2013). https://doi.org/10.1007/s10107-011-0488-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Hooker, J.N.: Logic, optimization, and constraint programming. INFORMS J. Comput. 14(4), 295–321 (2002). https://doi.org/10.1287/ijoc.14.4.295.2828

    Article  MathSciNet  MATH  Google Scholar 

  23. Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17(4), 851–868 (1996). https://doi.org/10.1137/S0895479894273134

    Article  MathSciNet  MATH  Google Scholar 

  24. Kanzow, C., Mehlitz, P., Steck, D.: Relaxation schemes for mathematical programmes with switching constraints. Optim. Methods Softw. (2019). https://doi.org/10.1080/10556788.2019.1663425

    Article  Google Scholar 

  25. Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. SIAM J. Optim. 23(2), 770–798 (2013). https://doi.org/10.1137/100802487

    Article  MathSciNet  MATH  Google Scholar 

  26. Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94(1), 115–135 (1997). https://doi.org/10.1023/A:1022659603268

    Article  MathSciNet  MATH  Google Scholar 

  27. Leyffer, S.: Complementarity constraints as nonlinear equations: theory and numerical experience. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings: Theory, Applications, and Algorithms, pp. 169–208. Springer, Boston (2006). https://doi.org/10.1007/0-387-34221-4_9

    Chapter  MATH  Google Scholar 

  28. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  29. Mehlitz, P.: On the linear independence constraint qualification in disjunctive programming. Optimization (2019). https://doi.org/10.1080/02331934.2019.1679811

    Article  Google Scholar 

  30. Mehlitz, P.: Stationarity conditions and constraint qualifications for mathematical programs with switching constraints. Math. Program. (2019). https://doi.org/10.1007/s10107-019-01380-5

    Article  Google Scholar 

  31. Mordukhovich, B.: Variational Analysis and Generalized Differentiation. Springer, Berlin (2006)

    Book  Google Scholar 

  32. Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic, Dordrecht (1998)

    Book  Google Scholar 

  33. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)

    Google Scholar 

  34. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001). https://doi.org/10.1137/S1052623499361233

    Article  MathSciNet  MATH  Google Scholar 

  35. Steffensen, S., Ulbrich, M.: A new relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 20(5), 2504–2539 (2010). https://doi.org/10.1137/090748883

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl. 13(1), 201–220 (1999). https://doi.org/10.1023/A:1008669226453

    Article  MathSciNet  MATH  Google Scholar 

  37. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Vieweg, Wiesbaden (2009)

    MATH  Google Scholar 

  38. Vinter, R.: Optimal Control. Birkhäuser, New York (2000)

    MATH  Google Scholar 

  39. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006). https://doi.org/10.1007/s10107-004-0559-y

    Article  MathSciNet  MATH  Google Scholar 

  40. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307(1), 350–369 (2005). https://doi.org/10.1016/j.jmaa.2004.10.032

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Brandenburgische Technische Universität Cottbus–Senftenberg, 03046, Cottbus, Germany

    Patrick Mehlitz

Authors
  1. Patrick Mehlitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick Mehlitz.

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

Mehlitz, P. A comparison of solution approaches for the numerical treatment of or-constrained optimization problems. Comput Optim Appl 76, 233–275 (2020). https://doi.org/10.1007/s10589-020-00169-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-020-00169-z

Keywords

Mathematics Subject Classification

Search

Navigation