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Tighter McCormick relaxations through subgradient propagation

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Abstract

Tight convex and concave relaxations are of high importance in deterministic global optimization. We present a method to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al. in SIAM J Optim 20(2):573–601, 2009) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the method and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the MINLPLib2 library and case studies presented in previous works, where the McCormick technique appears to be advantageous, and discuss computational efficiency. We see that the presented algorithm provides a significant improvement in tightness and decrease in computational time, especially in the case studies using the reduced space formulation presented in (Bongartz and Mitsos in J Glob Optim 69:761–796, 2017).

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References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bendtsen, C., Staunin, O.: FADBAD++, A Flexible C++ Package for Automatic Differentiation. Version 2.1 (2012). http://www.fadbad.com. Accessed 18 Octob 2016

  4. Bertsekas, D.P.: Convex Optimization Algorithms. Athena Scientific, Belmont (2015)

    MATH  Google Scholar 

  5. Bertsekas, D.P., Nedic, A., Ozdaglar, A.E., et al.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  6. Bompadre, A., Mitsos, A., Chachuat, B.: Convergence analysis of Taylor models and McCormick–Taylor models. J. Glob. Optim. 57(1), 75–114 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bongartz, D., Mitsos, A.: Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations. J. Glob. Optim. 69, 761–796 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bongartz, D., Mitsos, A.: Deterministic global flowsheet optimization - between equation-oriented and sequential-modular. AIChE J 65, 1022–1034 (2019)

    Article  Google Scholar 

  9. Bongartz, D., Najman, J., Sass, S., Mitsos, A.: MAiNGO: McCormick-based algorithm for mixed-integer nonlinear global optimization. In: Technical Report, Process Systems Engineering (AVT.SVT), RWTH Aachen University (2018). http://permalink.avt.rwth-aachen.de/?id=729717. Accessed 7 Jan 2019

  10. Brearley, A.L., Mitra, G., Williams, H.P.: Analysis of mathematical programming problems prior to applying the simplex algorithm. Math. Program. 8(1), 54–83 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chachuat, B., Houska, B., Paulen, R., Peri’c, N., Rajyaguru, J., Villanueva, M.E.: Set-theoretic approaches in analysis, estimation and control of nonlinear systems. IFAC PapersOnLine 48(8), 981–995 (2015)

    Article  Google Scholar 

  12. 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)

  13. Cornelius, H., Lohner, R.: Computing the range of values of real functions with accuracy higher than second order. Computing 33(3–4), 331–347 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. International Business Machines Corporation:: IBM ILOG CPLEX v12.8. Armonk (2009)

  15. De Figueiredo, L.H., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37(1–4), 147–158 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Gleixner, A.M., Berthold, T., Müller, B., Weltge, S.: Three enhancements for optimization-based bound tightening. J. Glob. Optim. 67(4), 731–757 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gould, N., Scott, J.: A note on performance profiles for benchmarking software. ACM Trans. Math. Softw. 43(2), 15:1–15:5 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hamed, A.S.E.D., McCormick, G.P.: Calculation of bounds on variables satisfying nonlinear inequality constraints. J. Glob. Optim. 3(1), 25–47 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hansen, P., Jaumard, B., Lu, S.H.: An analytical approach to global optimization. Math. Program. 52(1), 227–254 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, S.: The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/nlopt. Accessed Feb 2018

  22. Kannan, R., Barton, P.I.: The cluster problem in constrained global optimization. J. Glob. Optim. 69, 629–676 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kannan, R., Barton, P.I.: Convergence-order analysis of branch-and-bound algorithms for constrained problems. J. Glob. Optim. 71, 753–813 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kearfott, B., Du, K.: The Cluster Problem in Global Optimization: The Univariate Case, pp. 117–127. Springer, Vienna (1993)

    MATH  Google Scholar 

  25. Khan, K.A.: Subtangent-based approaches for dynamic set propagation. In: 57th IEEE Conference on Decision and Control (2018)

  26. Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24(4–5), 657–668 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Locatelli, M., Schoen, F.: Global optimization: theory, algorithms, and applications. In: SIAM, (2013)

  28. 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 

  29. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  31. Misener, R., Floudas, C.: Antigone: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Moore, R.E., Bierbaum, F.: Methods and applications of interval analysis. In: SIAM Studies in Applied and Numerical Mathematics, Society for Industrial and Applied Mathematics (1979)

  34. Morrison, D., Jacobson, S., Sauppe, J., Sewell, E.: Branch-and-bound algorithms: a survey of recent advances in searching, branching, and pruning. Discret. Optim. 19, 79–102 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Najman, J., Bongartz, D., Tsoukalas, A., Mitsos, A.: Erratum to: multivariate McCormick relaxations. J. Glob. Optim. 68, 1–7 (2016)

    MATH  Google Scholar 

  36. 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, Amsterdam (2016)

    Google Scholar 

  37. Najman, J., Mitsos, A.: On tightness and anchoring of McCormick and other relaxations. J. Glob. Optim. (2017). https://doi.org/10.1007/s10898-017-0598-6

  38. Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR 13(3), 247–277 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Puranik, Y., Sahinidis, N.V.: Bounds tightening based on optimality conditions for nonconvex box-constrained optimization. J. Glob. Optim. 67(1–2), 59–77 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Puranik, Y., Sahinidis, N.V.: Domain reduction techniques for global NLP and MINLP optimization. Constraints 22, 1–39 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. Halsted Press Chichester, New York (1984)

    MATH  Google Scholar 

  42. Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19(5), 551–566 (1995)

    Article  Google Scholar 

  43. Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sahlodin, A., Chachuat, B.: Convex, concave relaxations of parametric odes using taylor models. Comput. Chem. Eng. 35(5), 844–857, : Selected Papers from ESCAPE-20 (European Symposium of Computer Aided Process Engineering-20), Ischia, Italy (2011)

  45. Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33(4), 541–562 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Schweidtmann, A.M., Mitsos, A.: Global deterministic optimization with artificial neural networks embedded. J. Optim. Theory Appl. (2018)

  47. Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51(4), 569–606 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Shectman, J.P., Sahinidis, N.V.: A finite algorithm for global minimization of separable concave programs. J. Glob. Optim. 12(1), 1–36 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Smith, E.M., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, 791–796 (1997)

    Article  Google Scholar 

  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, New York (2002)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Glob. Optim. 59, 633–662 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Vigerske, S., Gleixner, A.: SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optim. Method. Softw. 33(3), 563–593 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wechsung, A.: Global optimization in reduced space. In: Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge (2014)

  56. Wechsung, A., Schaber, S.D., Barton, P.I.: The cluster problem revisited. J. Glob. Optim. 58(3), 429–438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wechsung, A., Scott, J.K., Watson, H.A.J., Barton, P.I.: Reverse propagation of McCormick relaxations. J. Glob. Optim. 63(1), 1–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Dr. Benoît Chachuat for providing MC++ v2.0 and Dominik Bongartz and Artur Schweidtmann for providing numerical case studies. We appreciate the thorough review and helpful comments provided by the anonymous reviewers and editors which resulted in an 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.

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  1. AVT - Aachener Verfahrenstechnik, Process Systems Engineering, RWTH Aachen University, Forckenbeckstraße 51, 52074, Aachen, Germany

    Jaromił Najman & Alexander Mitsos

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  1. Jaromił Najman
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A Appendix

A Appendix

See Tables 3, 4, 5 and 6.

Table 3 The problems for the numerical studies for different numbers of iterations within the heuristic
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Table 4 The numerical results. We allowed only 1 iteration within the heuristic and used the middle point of each node as its initial point
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Table 5 The numerical results. We allowed only 1 iteration within the heuristic and used the incumbent as its initial point
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Table 6 The numerical results for different numbers of iterations and points within the heuristic
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Najman, J., Mitsos, A. Tighter McCormick relaxations through subgradient propagation. J Glob Optim 75, 565–593 (2019). https://doi.org/10.1007/s10898-019-00791-0

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