Global optimization of MIQCPs with dynamic piecewise relaxations

Abstract

We propose a new deterministic global optimization algorithm for solving mixed-integer bilinear programs. It relies on a two-stage decomposition strategy featuring mixed-integer linear programming relaxations to compute estimates of the global optimum, and constrained non-linear versions of the original non-convex mixed-integer nonlinear program to find feasible solutions. As an alternative to spatial branch-and-bound with bilinear envelopes, we use extensively piecewise relaxations for computing estimates and reducing variable domain through optimality-based bound tightening. The novelty is that the number of partitions, a critical tuning parameter affecting the quality of the relaxation and computational time, increases and decreases dynamically based on the computational requirements of the previous iteration. Specifically, the algorithm alternates between piecewise McCormick and normalized multiparametric disaggregation. When solving ten benchmark problems from the literature, we obtain the same or better optimality gaps than two commercial global optimization solvers.

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References

  1. 1.

    Meyer, C.A., Floudas, C.A.: Global optimization of a combinatorially complex generalized pooling problem. AIChE J. 52, 1027–1037 (2006)

    Article  Google Scholar 

  2. 2.

    Misener, R., Thompson, J.P., Floudas, C.A.: APOGEE: global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput. Chem. Eng. 35, 876–892 (2011)

    Article  Google Scholar 

  3. 3.

    Castro, P.M.: New MINLP formulation for the multiperiod pooling problem. AIChE J. 61, 3728–3738 (2015)

    Article  Google Scholar 

  4. 4.

    Lotero, I., Trespalacios, F., Grossmann, I.E., Papageorgiou, D.J., Cheon, M.-S.: An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem. Comput. Chem. Eng. 87, 13–35 (2016)

    Article  Google Scholar 

  5. 5.

    Quesada, I., Grossmann, I.E.: Global optimization of bilinear process networks with multicomponent flows. Comput. Chem. Eng. 19, 1219–1242 (1995)

    Article  Google Scholar 

  6. 6.

    Lee, S., Grossmann, I.E.: Global optimization of nonlinear generalized disjunctive programming with bilinear equality constraints: applications to process networks. Comput. Chem. Eng. 27, 1557–1575 (2003)

    Article  Google Scholar 

  7. 7.

    Faria, D.C., Bagajewicz, M.J.: Novel bound contraction procedure for global optimization of bilinear MINLP problems with applications to water management problems. Comput. Chem. Eng. 35, 446–455 (2011)

    Article  Google Scholar 

  8. 8.

    Rubio-Castro, E., Ponce-Ortega, J.M., Serna-González, M., El-Halwagi, M.M., Pham, V.: Global optimization in property-based interplant water integration. AIChE J. 59, 813–833 (2013)

    Article  Google Scholar 

  9. 9.

    Alnouri, S., Linke, P., El-Halwagi, M.M.: Spatially constrained interplant water network synthesis with water treatment options. In: Eden, M.R., Siirola, J.D.S., Towler, G.P. (eds.) Proceedings of the 8th International Conference on Foundations of Computer-Aided Process Design, pp. 237–242. Elsevier, Amsterdam (2014)

    Google Scholar 

  10. 10.

    Teles, J.P., Castro, P.M., Matos, H.A.: Global optimization of water networks design using multiparametric disaggregation. Comput. Chem. Eng. 40, 132–147 (2012)

    Article  Google Scholar 

  11. 11.

    Koleva, M.N., Styan, C.A., Papageorgiou, L.G.: Optimisation approaches for the synthesis of water treatment plants. Comput. Chem. Eng. (2017)

  12. 12.

    Andrade, T., Ribas, G., Oliveira, F.: A strategy based on convex relaxation for solving the oil refinery operations planning problem. Ind. Eng. Chem. Res. 55, 144–155 (2016)

    Article  Google Scholar 

  13. 13.

    Castillo Castillo, P., Castro, P.M., Mahalec, V.: Global optimization algorithm for large-scale refinery planning models with bilinear terms. Ind. Eng. Chem. Res. 56, 530–548 (2017)

    Article  Google Scholar 

  14. 14.

    Castro, P.M., Grossmann, I.E.: Global optimal scheduling of crude oil blending operations with RTN continuous-time and multiparametric disaggregation. Ind. Eng. Chem. Res. 53, 15127–15145 (2014)

    Article  Google Scholar 

  15. 15.

    Cerdá, J., Pautasso, P.C., Cafaro, D.C.: Efficient approach for scheduling crude oil operations in marine-access refineries. Ind. Eng. Chem. Res. 54, 8219–8238 (2015)

    Article  Google Scholar 

  16. 16.

    Zhao, Y., Wu, N., Li, Z., Qu, T.: A novel solution approach to a priority-slot-based continuous-time mixed integer nonlinear programming formulation for a crude-oil scheduling problem. Ind. Eng. Chem. Res. 55, 10955–10967 (2016)

    Article  Google Scholar 

  17. 17.

    Catalão, J.P.S., Pousinho, H.M.I., Mendes, V.M.F.: Hydro energy systems management in Portugal: profit-based evaluation of a mixed-integer nonlinear approach. Energy 36, 500–507 (2011)

    Article  Google Scholar 

  18. 18.

    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (2013)

    Google Scholar 

  19. 19.

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

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Smith, E.M.B., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, S791–S796 (1997)

    Article  Google Scholar 

  21. 21.

    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 

  22. 22.

    Karuppiah, R., Grossmann, I.E.: Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 30, 650–673 (2006)

    Article  Google Scholar 

  23. 23.

    Alfaki, M., Haugland, D.: A multi-commodity flow formulation for the generalized pooling problem. J. Glob. Optim. 56, 917–937 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Bergamini, M.L., Aguirre, P., Grossmann, I.: Logic-based outer approximation for globally optimal synthesis of process networks. Comput. Chem. Eng. 29, 1914–1933 (2005)

    Article  Google Scholar 

  25. 25.

    Wicaksono, D.S., Karimi, I.A.: Piecewise MILP under- and overestimators for global optimization of bilinear programs. AIChE J. 54, 991–1008 (2008)

    Article  Google Scholar 

  26. 26.

    Li, X., Chen, Y., Barton, P.I.: Nonconvex generalized benders decomposition with piecewise convex relaxations for global optimization of integrated process design and operation problems. Ind. Eng. Chem. Res. 51, 7287–7299 (2012)

    Article  Google Scholar 

  27. 27.

    Castro, P.M.: Tightening piecewise McCormick relaxations for bilinear problems. Comput. Chem. Eng. 72, 300–311 (2015)

    Article  Google Scholar 

  28. 28.

    Kolodziej, S., Castro, P.M., Grossmann, I.E.: Global optimization of bilinear programs with a multiparametric disaggregation technique. J. Glob. Optim. 57, 1039–1063 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Castro, P.M.: Normalized multiparametric disaggregation: an efficient relaxation for mixed-integer bilinear problems. J. Glob. Optim. 64, 765–784 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Faria, D.C., Bagajewicz, M.J.: A new approach for global optimization of a class of MINLP problems with applications to water management and pooling problems. AIChE J. 58, 2320–2335 (2012)

    Article  Google Scholar 

  31. 31.

    Castro, P.M.: Spatial branch-and-bound algorithm for MIQCPs featuring multiparametric disaggregation. Optim. Methods Softw. 32, 719–737 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Castro, P.M., Teles, J.P.: Comparison of global optimization algorithms for the design of water-using networks. Comput. Chem. Eng. 52, 249–261 (2013)

    Article  Google Scholar 

  33. 33.

    Castro, P.M., Grossmann, I.E.: Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems. J. Glob. Optim. 59, 277–306 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Nagarajan, H., Lu, M., Yamangil, E., Bent, R.: Tightening McCormick relaxations for nonlinear programs via dynamic multivariate partitioning. In: Rueher, M. (ed.) Principles and Practice of Constraint Programming: 22nd International Conference, CP 2016, Toulouse, France, September 5–9, 2016, Proceedings, pp. 369–387. Springer, Cham (2016)

    Google Scholar 

  35. 35.

    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124, 383–411 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130, 359–413 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

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

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. 57, 3–50 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

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

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

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

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Atamturk, A., Nemhauser, G.L., Savelsbergh, M.W.P.: Conflict graphs in solving integer programming problems. Eur. J. Oper. Res. 121, 40–55 (2000)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

Support by Ontario Research Foundation, McMaster Advanced Control Consortium, and Fundação para a Ciência e Tecnologia (Projects IF/00781/2013 and UID/MAT/04561/2013), is gratefully appreciated.

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Correspondence to Pedro M. Castro.

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Castillo Castillo, P.A., Castro, P.M. & Mahalec, V. Global optimization of MIQCPs with dynamic piecewise relaxations. J Glob Optim 71, 691–716 (2018). https://doi.org/10.1007/s10898-018-0612-7

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Keywords

  • Mixed-integer nonlinear programming
  • Global optimization of quadratic programs with bilinear terms
  • Piecewise linear relaxations
  • Optimality-based bound tightening