Journal of Global Optimization

, Volume 74, Issue 4, pp 677–703 | Cite as

On tightness and anchoring of McCormick and other relaxations

  • Jaromił Najman
  • Alexander MitsosEmail author


We say that a convex relaxation of a function is anchored at a particular point in their domains if the values of the function and the relaxation at this point are equal. The opposite of anchoring is offset, i.e., a positive difference between the function and its convex relaxation values over the entire domain. We present theoretical results supported by theoretical and numerical examples showing that anchoring (at corner points) is a useful property but neither necessary nor sufficient for favorable Hausdorff and pointwise convergence order of a relaxation-based bounding scheme. Next, we investigate the tightness and convergence behavior of McCormick relaxations in specific cases. McCormick relaxations have favorable convergence orders, but a positive offset may still slow down the convergence within a simple branch-and-bound algorithm. We demonstrate that use of tighter underlying interval extensions can help reduce the offset and accelerate convergence.


Global optimization Nonconvex optimization Convergence order Convex relaxation McCormick Interval analysis 

Mathematics Subject Classification

49M20 49M37 65K05 90C26 



We would like to thank the late Prof. Floudas who motivated us to look into anchoring. We appreciate the thorough review and helpful comments provided by the anonymous reviewers and editors which resulted in a significantly improved manuscript. This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Improved McCormick Relaxations for the efficient Global Optimization in the Space of Degrees of Freedom MA 1188/34-1.


  1. 1.
    Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs-II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)CrossRefGoogle Scholar
  2. 2.
    Adjiman, C.S., Floudas, C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Global Optim. 9(1), 23–40 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: A global optimization method for general constrained nonconvex problems. J. Global Optim. 7(4), 337–363 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bao, X., Khajavirad, A., Sahinidis, N.V., Tawarmalani, M.: Global optimization of nonconvex problems with multilinear intermediates. Math. Program. Comput. 7(1), 1–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Global Optim. 52(1), 1–28 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bompadre, A., Mitsos, A., Chachuat, B.: Convergence analysis of Taylor models and McCormick-Taylor models. J. Global Optim. 57(1), 75–114 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bücker, M., Corliss, G., Hovland, P., Naumann, U., Norris, B.: Automatic Differentiation: Applications, Theory and Tools, vol. 50. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chachuat, B.: MC++: A versatile library for bounding and relaxation of factorable functions (2013). (February 2017), 2017)
  10. 10.
    Chachuat, B., Houska, B., Paulen, R., Perić, N., Rajyaguru, J., Villanueva, M.E.: Set-theoretic approaches in analysis, estimation and control of nonlinear systems. IFAC-PapersOnLine 48(8), 981–995 (2015). URL
  11. 11.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: Proceedings of VI SIBGRAPI (Brazilian Symposium on Computer Graphics and Image Processing), pp. 9–18. Citeseer (1993)Google Scholar
  12. 12.
    De Figueiredo, L.H., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Du, K., Kearfott, R.B.: The cluster problem in multivariate global optimization. J. Global Optim. 5(3), 253–265 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, vol. 1. Springer, Berlin (2008)zbMATHGoogle Scholar
  15. 15.
    Gatzke, E.P., Tolsma, J.E., Barton, P.I.: Construction of convex relaxations using automated code generation techniques. Optim. Eng. 3(3), 305–326 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Johnson, S.G.: The NLopt nonlinear-optimization package. 2017)
  18. 18.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kannan, R., Barton, P.I.: The cluster problem in constrained global optimization. J. Global Optim. (2017).
  20. 20.
    Kazazakis, N., Adjiman, C.S.: Globie: An algorithm for the deterministic global optimization of box-constrained NLPs. In: Eden, J.D.S. Mario R., Towler, G.P. (eds.) Proceedings of the 8th International Conference on Foundations of Computer-Aided Process Design. Computer Aided Chemical Engineering, vol. 34, pp. 669 – 674. Elsevier (2014). URL
  21. 21.
    Khajavirad, A., Michalek, J.J., Sahinidis, N.V.: Relaxations of factorable functions with convex-transformable intermediates. Math. Program. 144(1–2), 107–140 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khajavirad, A., Sahinidis, N.V.: Convex envelopes generated from finitely many compact convex sets. Math. Program. 137(1–2), 371–408 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Global Optim. 25(2), 157–168 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Locatelli, M., Schoen, F.: On convex envelopes for bivariate functions over polytopes. Math. Program. 144(1–2), 65–91 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Maranas, C.D., Floudas, C.A.: A global optimization approach for Lennard-Jones microclusters. The Journal of Chemical Physics 97(10), 7667–7678 (1992)CrossRefGoogle Scholar
  26. 26.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)CrossRefzbMATHGoogle Scholar
  27. 27.
    McCormick, G.P.: Nonlinear Programming: Theory, Algorithms, and Applications. Wiley, New York (1983)zbMATHGoogle Scholar
  28. 28.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103(2), 207–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Misener, R., Floudas, C.: Antigone: algorithms for continuous / integer global optimization of nonlinear equations. J. Global Optim. 59(2–3), 503–526 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mitsos, A., Chachuat, B., Barton, P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573–601 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mladineo, R.H.: An algorithm for finding the global maximum of a multimodal, multivariate function. Math. Program. 34(2), 188–200 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Moore, R.E., Bierbaum, F.: Methods and applications of interval analysis (SIAM Studies in Applied and Numerical Mathematics). Society for Industrial & Applied Math (1979)Google Scholar
  33. 33.
    Najman, J., Bongartz, D., Tsoukalas, A., Mitsos, A.: Erratum to: Multivariate McCormick relaxations. J. Global Optim. 1–7 (2016).
  34. 34.
    Najman, J., Mitsos, A.: Convergence analysis of multivariate McCormick relaxations. J Global Optim. 66(4), 597–628 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Najman, J., Mitsos, A.: Convergence order of McCormick relaxations of LMTD function in heat exchanger networks. In: Kravanja, Z., Bogataj, M. (eds.) 26th European Symposium on Computer Aided Process Engineering. Computer Aided Chemical Engineering, vol. 38, pp. 1605 – 1610. Elsevier (2016). URL
  36. 36.
    Neumaier, A.: Interval Methods for Systems of equations, vol. 37. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  37. 37.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR 13(3), 247–277 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pintér, J.: Extended univariate algorithms for n-dimensional global optimization. Computing 36(1), 91–103 (1986). MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, vol. 6. Springer, Berlin (2013)Google Scholar
  40. 40.
    Piyavskii, S.: An algorithm for finding the absolute extremum of a function. USSR Comput. Math. Math. Phys. 12(4), 57–67 (1972)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. E. Horwood; Halsted Press Chichester, New York (1984)zbMATHGoogle Scholar
  42. 42.
    Sahlodin, A., Chachuat, B.: Convex/concave relaxations of parametric odes using taylor models. Computers & Chemical Engineering 35(5), 844 – 857 (2011) URL Selected Papers from ESCAPE-20 (European Symposium of Computer Aided Process Engineering—20), 6–9 June 2010, Ischia, Italy
  43. 43.
    Schöbel, A., Scholz, D.: The theoretical and empirical rate of convergence for geometric branch-and-bound methods. J. Global Optim. 48(3), 473–495 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Global Optim. 51(4), 569–606 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V.: Benchmarking Global Optimization and Constraint Satisfaction Codes, pp. 211–222. Springer, Berlin (2003). CrossRefzbMATHGoogle Scholar
  46. 46.
    Smith, E.M., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, 791–796 (1997)CrossRefGoogle Scholar
  47. 47.
    Strongin, R.G.: Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves. J. Global Optim. 2(4), 357–378 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Global Optim. 20(2), 133–154 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Math. Program. 93, 247–263 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications, vol. 65. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  51. 51.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Global Optim. 59, 633–662 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vanderbei, R.J.: Extension of Piyavskii’s algorithm to continuous global optimization. J. Global Optim. 14(2), 205–216 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wechsung, A., Barton, P.I.: Global optimization of bounded factorable functions with discontinuities. J. Global Optim. 58(1), 1–30 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Wechsung, A., Schaber, S.D., Barton, P.I.: The cluster problem revisited. J. Global Optim. 58(3), 429–438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Zamora, J.M., Grossmann, I.E.: A global MINLP optimization algorithm for the synthesis of heat exchanger networks with no stream splits. Comput. Chem. Eng. 22(3), 367–384 (1998)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.RWTH Aachen University, AVT - Aachener Verfahrenstechnik, Process Systems EngineeringAachenGermany

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