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Relaxations and discretizations for the pooling problem

Abstract

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.

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Notes

  1. For the p-formulation, specifications serve the role of commodities and (4a) is a commodity balance constraint.

  2. For convenience, an equation that is satisfied by all feasible points in a set is also referred to as a valid inequality.

  3. Some generalized instances can also be found in Alfaki and Haugland [4] but in our experience the pq-formulations of these instances were solved by \(\texttt {BARON}\) in less than 15 min and hence seem to be relatively ease.

  4. Amongst the various choices for discretizing \(\mathbb {P}\), flow discretization was by far the best choice but the solutions from solving \({{\mathrm{\mathcal {B}}}}(\mathbb {F}\mathbb {P})\) were still very poor in comparison to discretizing \(\mathbb {PQ}\).

References

  1. Adhya, N., Tawarmalani, M., Sahinidis, N.: A Lagrangian approach to the pooling problem. Ind. Eng. Chem. Res. 38(5), 1956–1972 (1999)

    Article  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  3. Alfaki, M., Haugland, D.: A cost minimization heuristic for the pooling problem. Ann. Oper. Res. 222(1), 73–87 (2013a)

    MathSciNet  Article  MATH  Google Scholar 

  4. Alfaki, M., Haugland, D.: A multi-commodity flow formulation for the generalized pooling problem. J. Glob. Optim. 56(3), 917–937 (2013b)

    MathSciNet  Article  MATH  Google Scholar 

  5. Alfaki, M., Haugland, D.: Strong formulations for the pooling problem. J. Glob. Optim. 56(3), 897–916 (2013c)

    MathSciNet  Article  MATH  Google Scholar 

  6. Almutairi, H., Elhedhli, S.: A new Lagrangean approach to the pooling problem. J. Glob. Optim. 45(2), 237–257 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  7. Audet, C., Brimberg, J., Hansen, P., Le Digabel, S., Mladenović, N.: Pooling problem: alternate formulations and solution methods. Manag. Sci. 50(6), 761–776 (2004)

    Article  MATH  Google Scholar 

  8. Audet, C., Hansen, P., Jaumard, B., Savard, G.: A symmetrical linear maxmin approach to disjoint bilinear programming. Math. Program. 85(3), 573–592 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  9. Baker, T., Lasdon, L.: Successive linear programming at Exxon. Manag. Sci. 31(3), 264–274 (1985)

    Article  MATH  Google Scholar 

  10. Bao, X., Sahinidis, N., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Methods Softw. 24(4–5), 485–504 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  11. Bao, X., Sahinidis, N., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129(1), 129–157 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  12. Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  13. Ben-Tal, A., Eiger, G., Gershovitz, V.: Global minimization by reducing the duality gap. Math. Program. 63(1), 193–212 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  14. Biegler, L., Grossmann, I., Westerberg, A.: Systematic methods for chemical process design. In: International Series in the Physical and Chemical Engineering Sciences. Prentice Hall (1997)

  15. Bley, A., Boland, N., Froyland, G., Zuckerberg, M.: Solving mixed integer nonlinear programming problems for mine production planning with stockpiling (2012). http://www.optimization-online.org/DB_HTML/2012/11/3674.html

  16. Bodington, C., Baker, T.: A history of mathematical programming in the petroleum industry. Interfaces 20(4), 117–127 (1990)

    Article  Google Scholar 

  17. Boland, N., Kalinowski, T., Rigterink, F.: New multi-commodity flow formulations for the pooling problem. J. Glob. Optim (2015). doi:10.1007/s10898-016-0404-x

  18. Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)

    MathSciNet  Google Scholar 

  19. Burer, S., Saxena, A.: The MILP road to MIQCP. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 373–405. Springer, Berlin (2012)

    Chapter  Google Scholar 

  20. Crama, Y.: Concave extensions for nonlinear 0–1 maximization problems. Math. Program. 61(1–3), 53–60 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  21. D’Ambrosio, C., Linderoth, J., Luedtke, J.: Valid inequalities for the pooling problem with binary variables. In: Günlük, O., Woeginger, G. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 6655, pp. 117–129. Springer (2011)

  22. Dey, S., Gupte, A.: Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2), 412–427 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  23. Floudas, C., Aggarwal, A.: A decomposition strategy for global optimum search in the pooling problem. ORSA J. Comput. 2(3), 225–235 (1990)

    Article  MATH  Google Scholar 

  24. Foulds, L., Haugland, D., Jörnsten, K.: A bilinear approach to the pooling problem. Optimization 24(1), 165–180 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  25. Frimannslund, L., El Ghami, M., Alfaki, M., Haugland, D.: Solving the pooling problem with LMI relaxations. In: TOGO10—global optimization workshop, pp. 51–54 (2010)

  26. Frimannslund, L., Gundersen, G., Haugland, D.: Sensitivity analysis applied to the pooling problem. Tech. Rep. 380, University of Bergen (2008)

  27. Furman, K., Androulakis, I.: A novel MINLP-based representation of the original complex model for predicting gasoline emissions. Comp. Chem. Eng. 32(12), 2857–2876 (2008)

    Article  Google Scholar 

  28. Gounaris, C., Misener, R., Floudas, C.: Computational comparison of piecewise-linear relaxations for pooling problems. Ind. Eng. Chem. Res. 48(12), 5742–5766 (2009)

    Article  Google Scholar 

  29. Greenberg, H.: Analyzing the pooling problem. ORSA J. Comput. 7(2), 205–217 (1995)

    Article  MATH  Google Scholar 

  30. Günlük, O., Lee, J., Leung, J.: A polytope for a product of real linear functions in 0/1 variables. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 513–529. Springer, Berlin (2012)

    Chapter  Google Scholar 

  31. Gupte, A.: Mixed integer bilinear programming with applications to the pooling problem. Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA (2012). https://smartech.gatech.edu/handle/1853/45761

  32. Gupte, A.: Bilinear programming with simplicial constraints (2016a). Working paper. http://people.clemson.edu/~agupte/BilinSimpl.pdf

  33. Gupte, A.: Convex hulls of superincreasing knapsacks and lexicographic orderings. Discrete Appl. Math. 201, 150–163 (2016b)

    MathSciNet  Article  MATH  Google Scholar 

  34. Gupte, A., Ahmed, S., Cheon, M., Dey, S.: Solving mixed integer bilinear problems using MILP formulations. SIAM J. Optim. 23(2), 721–744 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  35. Hasan, M., Karimi, I.: Piecewise linear relaxation of bilinear programs using bivariate partitioning. AIChE J. 56(7), 1880–1893 (2010)

    Article  Google Scholar 

  36. Haugland, D.: The computational complexity of the pooling problem. J. Glob. Optim. 1–17 (2015). doi:10.1007/s10898-015-0335-y

  37. Haverly, C.: Studies of the behavior of recursion for the pooling problem. ACM SIGMAP Bull. 25, 19–28 (1978)

    Article  Google Scholar 

  38. Kallrath, J.: Solving planning and design problems in the process industry using mixed integer and global optimization. Ann. Oper. Res. 140(1), 339–373 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  39. Karuppiah, R., Furman, K., Grossmann, I.: Global optimization for scheduling refinery crude oil operations. Comput. Chem. Eng. 32(11), 2745–2766 (2008)

    Article  Google Scholar 

  40. Karuppiah, R., Grossmann, I.: Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 30(4), 650–673 (2006)

    Article  Google Scholar 

  41. Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15(3–4), 201–224 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  42. Kolodziej, S.P., Grossmann, I.E., Furman, K.C., Sawaya, N.W.: A discretization-based approach for the optimization of the multiperiod blend scheduling problem. Comput. Chem. Eng. 53, 122–142 (2013)

    Article  Google Scholar 

  43. Lee, S., Grossmann, I.: Global optimization of nonlinear generalized disjunctive programming with bilinear equality constraints: applications to process networks. Comput. Chem. Eng. 27(11), 1557–1575 (2003)

    Article  Google Scholar 

  44. Li, X., Armagan, E., Tomasgard, A., Barton, P.I.: Stochastic pooling problem for natural gas production network design and operation under uncertainty. AIChE J. 57(8), 2120–2135 (2011)

    Article  Google Scholar 

  45. Li, X., Tomasgard, A., Barton, P.I.: Decomposition strategy for the stochastic pooling problem. J. Glob. Optim. 54(4), 765–790 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  46. Liberti, L., Pantelides, C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim. 36(2), 161–189 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  47. Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Program. 136(2), 325–351 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  48. Marcotte, O.: The cutting stock problem and integer rounding. Math. Program. 33(1), 82–92 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  49. McCormick, G.: Computability of global solutions to factorable nonconvex programs: part I. convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    Article  MATH  Google Scholar 

  50. Meyer, C., Floudas, C.: Global optimization of a combinatorially complex generalized pooling problem. AIChE J. 52(3), 1027–1037 (2006)

    Article  Google Scholar 

  51. Misener, R., Floudas, C.: Advances for the pooling problem: modeling, global optimization, and computational studies. Appl. Comput. Math. 8(1), 3–22 (2009)

    MathSciNet  MATH  Google Scholar 

  52. Misener, R., Floudas, C.: Global optimization of large-scale generalized pooling problems: quadratically constrained MINLP models. Ind. Eng. Chem. Res. 49(11), 5424–5438 (2010)

    Article  Google Scholar 

  53. Misener, R., Gounaris, C., Floudas, C.: Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints. Comput. Chem. Eng. 34(9), 1432–1456 (2010)

    Article  Google Scholar 

  54. Misener, R., Smadbeck, J.B., Floudas, C.A.: Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2. Optim. Methods Softw. 30(1), 215–249 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  55. Misener, R., Thompson, J., Floudas, C.: 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 

  56. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization, Discrete Mathematics and Optimization, vol. 18. Wiley-Interscience, London (1988)

    MATH  Google Scholar 

  57. Nishi, T.: A semidefinite programming relaxation approach for the pooling problem. Master’s thesis, Department of Applied Mathematics and Physics, Kyoto University (2010). http://www-optima.amp.i.kyoto-u.ac.jp/result/masterdoc/21nishi.pdf

  58. Pham, V., Laird, C., El-Halwagi, M.: Convex hull discretization approach to the global optimization of pooling problems. Ind. Eng. Chem. Res. 48(4), 1973–1979 (2009)

    Article  Google Scholar 

  59. Quesada, I., Grossmann, I.: Global optimization of bilinear process networks with multicomponent flows. Comput. Chem. Eng. 19(12), 1219–1242 (1995)

    Article  Google Scholar 

  60. Realff, M., Ahmed, S., Inacio, H., Norwood, K.: Heuristics and upper bounds for a pooling problem with cubic constraints. In: Foundations of Computer-Aided Process Operations. Savannah, GA (2012). http://focapo.cheme.cmu.edu/2012/proceedings/data/papers/056.pdf

  61. Rikun, A.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10(4), 425–437 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  62. Ruiz, J., Grossmann, I.: Exploiting vector space properties to strengthen the relaxation of bilinear programs arising in the global optimization of process networks. Optim. Lett. 5(1), 1–11 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  63. Ruiz, M., Briant, O., Clochard, J., Penz, B.: Large-scale standard pooling problems with constrained pools and fixed demands. J. Glob. Optim. 56(3), 939–956 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  64. Sherali, H., Adams, W.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Nonconvex Optimization and its Applications, vol. 31. Kluwer Academic Publishers, Dordrecht (1998)

    Google Scholar 

  65. Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta Math. Vietnam. 22(1), 245–270 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  67. Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2(3), 363–375 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  68. Tawarmalani, M., Sahinidis, N.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  69. Vielma, J., Nemhauser, G.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128, 49–72 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  70. Visweswaran, V.: MINLP: applications in blending and pooling problems. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 2114–2121. Springer, Berlin (2009)

    Chapter  Google Scholar 

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Acknowledgments

We thank one referee whose elaborate comments helped us significantly improve the readibility of this paper. Megan Ryan, a graduate student at Clemson University, assisted the first author with the computational experiments in Sect. 5.1.

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Gupte, A., Ahmed, S., Dey, S.S. et al. Relaxations and discretizations for the pooling problem. J Glob Optim 67, 631–669 (2017). https://doi.org/10.1007/s10898-016-0434-4

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Keywords

  • Pooling problem
  • Bilinear program
  • Convexification
  • Lagrange relaxation
  • Discretization