Skip to main content
SpringerLink
Log in

Convergence rate of McCormick relaxations

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

Abstract

Theory for the convergence order of the convex relaxations by McCormick (Math Program 10(1):147–175, 1976) for factorable functions is developed. Convergence rules are established for the addition, multiplication and composition operations. The convergence order is considered both in terms of pointwise convergence and of convergence in the Hausdorff metric. The convergence order of the composite function depends on the convergence order of the relaxations of the factors. No improvement in the order of convergence compared to that of the underlying bound calculation, e.g., via interval extensions, can be guaranteed unless the relaxations of the factors have pointwise convergence of high order. The McCormick relaxations are compared with the αBB relaxations by Floudas and coworkers (J Chem Phys, 1992, J Glob Optim, 1995, 1996), which guarantee quadratic convergence. Illustrative and numerical examples are given.

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

Similar content being viewed by others

References

  1. Adjiman C.S., Androulakis I.P., Floudas C.A.: A global optimization method, αBB for general twice-differentiable constrained NLPs—II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)

    Article  Google Scholar 

  2. Adjiman C.S., Androulakis I.P., Maranas C.D., Floudas C.A.: A global optimization method, αBB for process design. Comput. Chem. Eng. 20(Suppl A), S419–S424 (1996)

    Article  Google Scholar 

  3. Adjiman C.S., Dallwig S., Floudas C.A., Neumaier A.: A global optimization method, αBB for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  4. Adjiman C.S., Floudas C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9(1), 23–40 (1996)

    Article  Google Scholar 

  5. Akrotirianakis I.G., Floudas C.A.: Computational experience with a new class of convex underestimators: Box-constrained NLP problems. J. Glob. Optim. 29(3), 249–264 (2004)

    Article  Google Scholar 

  6. Akrotirianakis I.G., Floudas C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30(4), 367–390 (2004)

    Article  Google Scholar 

  7. Al-Khayyal F.A., Falk J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)

    Article  Google Scholar 

  8. Alefeld G., Mayer G.: Interval analysis: Theory and applications. J. Comput. Appl. Math. 121(1–2), 421–464 (2000)

    Article  Google Scholar 

  9. Androulakis I.P., Maranas C.D., Floudas C.A.: αBB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995)

    Article  Google Scholar 

  10. Belotti P., Lee J., Liberti L., Margot F., Wachter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)

    Article  Google Scholar 

  11. Bhattacharjee B., Green W.H. Jr., Barton P.I.: Interval methods for semi-infinite programs. Comput. Optim. Appl. 30(1), 63–93 (2005)

    Article  Google Scholar 

  12. Bhattacharjee B., Lemonidis P., Green W.H. Jr., Barton P.I.: Global solution of semi-infinite programs. Math. Program. Ser. B 103(2), 283–307 (2005)

    Article  Google Scholar 

  13. Chachuat, B.: libMC: A numeric library for McCormick relaxation of factorable functions. Documentation and Source Code available at: http://yoric.mit.edu/libMC/ (2007)

  14. Chachuat, B.: MC++: A versatile library for McCormick relaxations and Taylor models. Documentation and Source Code available at: http://www3.imperial.ac.uk/people/b.chachuat/research (2010)

  15. Du K.S., Kearfott R.B.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5(3), 253–265 (1994)

    Article  Google Scholar 

  16. Gatzke E.P., Tolsma J.E., Barton P.I.: Construction of convex function relaxations using automated code generation techniques. Optim. Eng. 3(3), 305–326 (2002)

    Article  Google Scholar 

  17. Goldstein A.A., Price J.F.: Descent from local minima. Math. Comput. 25(115), 569–574 (1971)

    Article  Google Scholar 

  18. Gounaris C.E., Floudas C.A.: Tight convex underestimators for C-2-continuous problems: I. Univariate functions. J. Glob. Optim. 42(1), 51–67 (2008)

    Article  Google Scholar 

  19. Gounaris C.E., Floudas C.A.: Tight convex underestimators for C-2-continuous problems: II. Multivariate functions. J. Glob. Optim. 42(1), 69–89 (2008)

    Article  Google Scholar 

  20. Maranas C.D., Floudas C.A.: A global optimization approach for Lennard-Jones microclusters. J. Chem. Phys. 97(10), 7667–7678 (1992)

    Article  Google Scholar 

  21. Maranas C.D., Floudas C.A.: Global optimization for molecular conformation problems. Ann. Oper. Res. 42(3), 85–117 (1993)

    Article  Google Scholar 

  22. Maranas C.D., Floudas C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4, 135–170 (1994)

    Article  Google Scholar 

  23. Maranas C.D., Floudas C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143–182 (1995)

    Article  Google Scholar 

  24. McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I. Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    Article  Google Scholar 

  25. McCormick G.P.: Nonlinear Programming: Theory, Algorithms and Applications. Wiley, New York (1983)

    Google Scholar 

  26. Mitsos A., Chachuat B., Barton P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573–601 (2009)

    Article  Google Scholar 

  27. Mitsos A., Lemonidis P., Lee C.K., Barton P.I.: Relaxation-based bounds for semi-infinite programs. SIAM J. Optim. 19(1), 77–113 (2008)

    Article  Google Scholar 

  28. Moore R.: Methods and Applications of Interval Analysis. SIAM, Philadelphia, PA (1979)

    Book  Google Scholar 

  29. Munkres J.: Topology, 2nd edn. Prentice Hall, Englewood Cliffs (1999)

    Google Scholar 

  30. Neumaier A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13, 271–369 (2004)

    Article  Google Scholar 

  31. Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. Ellis Horwood Series, Mathematics and its Applications, New York (1984)

  32. Ryoo H.S., Sahinidis N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19(4), 403–424 (2001)

    Article  Google Scholar 

  33. Sahinidis, N., Tawarmalani, M.: BARON. http://www.gams.com/solvers/baron.pdf (2005)

  34. 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)

    Article  Google Scholar 

  35. Smith E.M.B., Pantelides C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23(4–5), 457–478 (1999)

    Article  Google Scholar 

  36. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Boston (2002)

  37. Tawarmalani M., Sahinidis N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program. 99(3), 563–591 (2004)

    Article  Google Scholar 

  38. Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    Article  Google Scholar 

  39. Zlobec S.: On the Liu-Floudas convexification of smooth programs. J. Glob. Optim. 32(3), 401–407 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA, 91125, USA

    Agustín Bompadre

  2. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Alexander Mitsos

Authors
  1. Agustín Bompadre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Mitsos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Mitsos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bompadre, A., Mitsos, A. Convergence rate of McCormick relaxations. J Glob Optim 52, 1–28 (2012). https://doi.org/10.1007/s10898-011-9685-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9685-2

Keywords

Search

Navigation