Journal of Global Optimization

, Volume 69, Issue 3, pp 629–676 | Cite as

The cluster problem in constrained global optimization

Article

Abstract

Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (Du and Kearfott in J Glob Optim 5(3):253–265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (Du and Kearfott 1994; Wechsung et al. in J Glob Optim 58(3):429–438, 2014) to the constrained setting based on a recently-developed notion of convergence order for convex relaxation-based lower bounding schemes. It is shown that clustering can occur both on nearly-optimal and nearly-feasible regions in the vicinity of a global minimizer. In contrast to the case of unconstrained optimization, where at least second-order convergent schemes of relaxations are required to mitigate the cluster problem when the minimizer sits at a point of differentiability of the objective function, it is shown that first-order convergent lower bounding schemes for constrained problems may mitigate the cluster problem under certain conditions. Additionally, conditions under which second-order convergent lower bounding schemes are sufficient to mitigate the cluster problem around a global minimizer are developed. Conditions on the convergence order prefactor that are sufficient to altogether eliminate the cluster problem are also provided. This analysis reduces to previous analyses of the cluster problem for unconstrained optimization under suitable assumptions.

Keywords

Cluster problem Global optimization Constrained optimization Branch-and-bound Convergence order Convex relaxation Lower bounding scheme 

Mathematics Subject Classification

49M20 49M37 65K05 68Q25 90C26 90C46 

Notes

Acknowledgements

The authors would like to thank three anonymous reviewers and an associate editor for suggestions which helped improve the readability of this article, and Dr. Johannes Jäschke for helpful discussions (in particular, for bring references [5] and [12] to our attention). The first author would also like to thank Jaichander Swaminathan for helpful discussions regarding the proof of Theorem 2.

References

  1. 1.
    Adjiman, C.S., Floudas, C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9(1), 23–40 (1996)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, Hoboken (2013)MATHGoogle Scholar
  3. 3.
    Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Glob. Optim. 52(1), 1–28 (2012)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bompadre, A., Mitsos, A., Chachuat, B.: Convergence analysis of Taylor models and McCormick–Taylor models. J. Glob. Optim. 57(1), 75–114 (2013)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bonnans, J.F., Ioffe, A.: Second-order sufficiency and quadratic growth for nonisolated minima. Math. Oper. Res. 20(4), 801–817 (1995)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Clarke, F.H.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)CrossRefGoogle Scholar
  7. 7.
    Du, K., Kearfott, R.B.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5(3), 253–265 (1994)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of test problems in local and global optimization. In: Nonconvex Optimization and Its Applications, 1st edn, vol. 33. Springer, Berlin (1999)Google Scholar
  9. 9.
    Goldsztejn, A., Domes, F., Chevalier, B.: First order rejection tests for multiple-objective optimization. J. Glob. Optim. 58(4), 653–672 (2014)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hijazi, H., Liberti, L.: Constraint qualification failure in action. Oper. Res. Lett. 44(4), 503–506 (2016)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  12. 12.
    Ioffe, A.: On sensitivity analysis of nonlinear programs in Banach spaces: the approach via composite unconstrained optimization. SIAM J. Optim. 4(1), 1–43 (1994)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kearfott, R.B., Du, K.: The cluster problem in global optimization: the univariate case. In: Validation Numerics, Computing Supplementum, vol. 9, pp. 117–127. Springer, Berlin (1993)Google Scholar
  14. 14.
    Khan, K.A., Watson, H.A.J., Barton, P.I.: Differentiable McCormick relaxations. J. Glob. Optim. 67(4), 687–729 (2017)Google Scholar
  15. 15.
    Krawczyk, R., Nickel, K.: Die zentrische form in der Intervallarithmetik, ihre quadratische Konvergenz und ihre Inklusionsisotonie. Computing 28(2), 117–137 (1982)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mayer, G.: Epsilon-inflation in verification algorithms. J. Comput. Appl. Math. 60(1), 147–169 (1995)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefMATHGoogle Scholar
  18. 18.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Najman, J., Bongartz, D., Tsoukalas, A., Mitsos, A.: Erratum to: multivariate McCormick relaxations. J. Glob. Optim. 68(1), 219–225 (2017)CrossRefMATHGoogle Scholar
  20. 20.
    Najman, J., Mitsos, A.: Convergence analysis of multivariate McCormick relaxations. J. Glob. Optim. 66(4), 597–628 (2016)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13, 271–369 (2004)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)MATHGoogle Scholar
  23. 23.
    Schöbel, A., Scholz, D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Glob. Optim. 48(3), 473–495 (2010)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. SpringerBriefs in Optimization, 1st edn. Springer, New York (2012)CrossRefMATHGoogle Scholar
  25. 25.
    Scholz, D.: Theoretical rate of convergence for interval inclusion functions. J. Glob. Optim. 53(4), 749–767 (2012)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Glob. Optim. 59(2–3), 633–662 (2014)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Van Iwaarden, R.J.: An improved unconstrained global optimization algorithm. Ph.D. thesis, University of Colorado at Denver (1996)Google Scholar
  28. 28.
    Wechsung, A.: Global optimization in reduced space. Ph.D. thesis, Massachusetts Institute of Technology (2014)Google Scholar
  29. 29.
    Wechsung, A., Schaber, S.D., Barton, P.I.: The cluster problem revisited. J. Glob. Optim. 58(3), 429–438 (2014)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Process Systems Engineering Laboratory, Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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