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Improved Convex and Concave Relaxations of Composite Bilinear Forms

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Abstract

Deterministic nonconvex optimization solvers generate convex relaxations of nonconvex functions by making use of underlying factorable representations. One approach introduces auxiliary variables assigned to each factor that lifts the problem into a higher-dimensional decision space. In contrast, a generalized McCormick relaxation approach offers the significant advantage of constructing relaxations in the lower dimensionality space of the original problem without introducing auxiliary variables, often referred to as a “reduced-space” approach. Recent contributions illustrated how additional nontrivial inequality constraints may be used in factorable programming to tighten relaxations of the ubiquitous bilinear term. In this work, we exploit an analogous representation of McCormick relaxations and factorable programming to formulate tighter relaxations in the original decision space. We develop the underlying theory to generate necessarily tighter reduced-space McCormick relaxations when a priori convex/concave relaxations are known for intermediate bilinear terms. We then show how these rules can be generalized within a McCormick relaxation scheme via three different approaches: the use of a McCormick relaxations coupled to affine arithmetic, the propagation of affine relaxations implied by subgradients, and an enumerative approach that directly uses relaxations of each factor. The developed approaches are benchmarked on a library of optimization problems using the EAGO.jl optimizer. Two case studies are also considered to demonstrate the developments: an application in advanced manufacturing to optimize supply chain quality metrics and a global dynamic optimization application for rigorous model validation of a kinetic mechanism. The presented subgradient method leads to an improvement in CPU time required to solve the considered problems to \(\epsilon \)-global optimality.

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Data Availability

The datasets generated during and/or analysed during the current study are available in a GitHub repository: https://github.com/PSORLab/RSBilinear.

References

  1. Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia (1999). https://doi.org/10.1137/1.9780898719604

    Book  MATH  Google Scholar 

  2. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Methods Softw. 24(4–5), 485–504 (2009). https://doi.org/10.1080/10556780902883184

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129(1), 129–157 (2011). https://doi.org/10.1007/s10107-011-0462-2

    Article  MathSciNet  MATH  Google Scholar 

  4. Benson, H.P.: Separable concave minimization via partial outer approximation and branch and bound. Oper. Res. Lett. 9(6), 389–394 (1990). https://doi.org/10.1016/0167-6377(90)90059-E

    Article  MathSciNet  MATH  Google Scholar 

  5. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017). https://doi.org/10.1137/141000671

    Article  MathSciNet  MATH  Google Scholar 

  6. Blackford, L.S., Demmel, J., Dongarra, J., Duff, I., Hammarling, S., Henry, G., Heroux, M., Kaufman, L., Lumsdaine, A., Petitet, A., Pozo, R., Remington, K., Whaley, R.C.: An updated set of basic linear algebra subprograms (BLAS). ACM Trans. Math. Softw. 28(2), 135–151 (2002). https://doi.org/10.1145/567806.567807

    Article  MathSciNet  Google Scholar 

  7. Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Global Optim. 52(1), 1–28 (2011). https://doi.org/10.1007/s10898-011-9685-2

    Article  MathSciNet  MATH  Google Scholar 

  8. Bongartz, D., Mitsos, A.: Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations. J. Global Optim. 69(4), 761–796 (2017). https://doi.org/10.1007/s10898-017-0547-4

    Article  MathSciNet  MATH  Google Scholar 

  9. Bongartz, D., Mitsos, A.: Deterministic global flowsheet optimization: between equation-oriented and sequential-modular methods. AIChE J. 65(3), 1022–1034 (2019). https://doi.org/10.1002/aic.16507

    Article  Google Scholar 

  10. Bongartz, D., Najman, J., Mitsos, A.: Deterministic global optimization of steam cycles using the IAPWS-IF97 model. Optim. Eng. (2020). https://doi.org/10.1007/s11081-020-09502-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Bongartz, D., Najman, J., Sass, S., Mitsos, A.: MAiNGO: McCormick based algorithm for mixed integer nonlinear global optimization. Technical report, RWTH Aachen (2018). https://git.rwth-aachen.de/avt-svt/public/maingo

  12. Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: de Figueiredo, L.H., de Miranda Gomes, J. (eds.) Proceedings of VI SIBGRAPI (Brazilian Symposium on Computer Graphics and Image Processing), pp. 9–18 (1993). http://urlib.net/ibi/8JMKD3MGPBW34M/3D6UQ68

  13. de Figueiredo, L.H., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37, 147–158 (2004). https://doi.org/10.1023/b:numa.0000049462.70970.b6

    Article  MathSciNet  MATH  Google Scholar 

  14. De Oliveira, L.G., de Paiva, A.P., Balestrassi, P.P., Ferreira, J.R., da Costa, S.C., da Silva Campos, P.H.: Response surface methodology for advanced manufacturing technology optimization: theoretical fundamentals, practical guidelines, and survey literature review. Int. J. Adv. Manuf. Technol. 104(5), 1785–1837 (2019). https://doi.org/10.1007/s00170-019-03809-9

    Article  Google Scholar 

  15. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002). https://doi.org/10.1007/s101070100263

    Article  MathSciNet  MATH  Google Scholar 

  16. Falk, J.E., Hoffman, K.L.: Concave minimization via collapsing polytopes. Oper. Res. 34(6), 919–929 (1986). https://doi.org/10.1287/opre.34.6.919

    Article  MathSciNet  MATH  Google Scholar 

  17. Fedorov, G., Nguyen, K.T., Harrison, P., Singh, A.: Intel Math Kernel Library 2019 Update 2 Release Notes (2019). https://software.intel.com/en-us/mkl

  18. Gurobi Optimization, LLC: Gurobi optimizer reference manual (2020). http://www.gurobi.com

  19. He, T., Tawarmalani, M.: A new framework to relax composite functions in nonlinear programs. Mathematical Programming, pp. 1–40 (2020). https://doi.org/10.1007/s10107-020-01541-x

  20. Hejazi, T.H., Seyyed-Esfahani, M., Mahootchi, M.: Quality chain design and optimization by multiple response surface methodology. Int. J. Adv. Manuf. Technol. 68(1–4), 881–893 (2013). https://doi.org/10.1007/s00170-013-4950-9

    Article  Google Scholar 

  21. Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-56468-0

    Book  MATH  Google Scholar 

  22. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996). https://doi.org/10.1007/978-3-662-03199-5

    Book  MATH  Google Scholar 

  23. IBM: IBM ILOG CPLEX Optimizer. https://www.ibm.com/software/integration/optimization/cplex-optimizer/ (2020)

  24. Kannan, R., Barton, P.I.: The cluster problem in constrained global optimization. J. Global Optim. 69(3), 629–676 (2017). https://doi.org/10.1007/s10898-017-0531-z

    Article  MathSciNet  MATH  Google Scholar 

  25. Khan, K.A., Watson, H.A.J., Barton, P.I.: Differentiable McCormick relaxations. J. Global Optim. 67(4), 687–729 (2016). https://doi.org/10.1007/s10898-016-0440-6

    Article  MathSciNet  MATH  Google Scholar 

  26. Khan, K.A., Wilhelm, M., Stuber, M.D., Cao, H., Watson, H.A.J., Barton, P.I.: Corrections to: Differentiable McCormick relaxations. J. Global Optim. 70(3), 705–706 (2018). https://doi.org/10.1007/s10898-017-0601-2

    Article  MathSciNet  MATH  Google Scholar 

  27. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976). https://doi.org/10.1007/bf01580665

    Article  MATH  Google Scholar 

  28. Messine, F.: Extensions of affine arithmetic: application to unconstrained global optimization. J. Univ. Comput. Sci. 8(11), 992–1015 (2002). https://doi.org/10.3217/jucs-008-11-0992

    Article  MathSciNet  MATH  Google Scholar 

  29. Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Global Optim. 57(1), 3–50 (2013). https://doi.org/10.1007/s10898-012-9874-7

    Article  MathSciNet  MATH  Google Scholar 

  30. Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Global Optim. 59(2–3), 503–526 (2014). https://doi.org/10.1007/s10898-014-0166-2

    Article  MathSciNet  MATH  Google Scholar 

  31. Mistry, M., Misener, R.: Optimising heat exchanger network synthesis using convexity properties of the logarithmic mean temperature difference. Comput. Chem. Eng. 94, 1–17 (2016). https://doi.org/10.1016/j.compchemeng.2016.07.001

    Article  Google Scholar 

  32. Mitsos, A., Chachuat, B., Barton, P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573–601 (2009). https://doi.org/10.1137/080717341

    Article  MathSciNet  MATH  Google Scholar 

  33. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ (1966)

    MATH  Google Scholar 

  34. Mosek, A.: The MOSEK optimization software (2010). https://www.mosek.com

  35. Mostaan, H., Shamanian, M., Safari, M.: Process analysis and optimization for fracture stress of electron beam welded ultra-thin FeCo-V foils. Int. J. Adv. Manuf. Technol. 87(1), 1045–1056 (2016). https://doi.org/10.1007/s00170-016-8553-0

    Article  Google Scholar 

  36. Nagarajan, H., Lu, M., Wang, S., Bent, R., Sundar, K.: An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs. J. Global Optim. (2019). https://doi.org/10.1007/s10898-018-00734-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Nagarajan, H., Lu, M., Yamangil, E., Bent, R.: Tightening McCormick relaxations for nonlinear programs via dynamic multivariate partitioning. In: Rueher, M. (ed.) International Conference on Principles and Practice of Constraint Programming, pp. 369–387. Springer, New York (2016). https://doi.org/10.1007/978-3-319-44953-1_24

    Chapter  Google Scholar 

  38. Najman, J., Bongartz, D., Mitsos, A.: Convex relaxations of componentwise convex functions. Comput. Chem. Eng. 130, 106527 (2019). https://doi.org/10.1016/j.compchemeng.2019.106527

    Article  Google Scholar 

  39. Najman, J., Bongartz, D., Mitsos, A.: Relaxations of thermodynamic property and costing models in process engineering. Comput. Chem. Eng. 130, 106571 (2019). https://doi.org/10.1016/j.compchemeng.2019.106571

    Article  Google Scholar 

  40. Najman, J., Bongartz, D., Tsoukalas, A., Mitsos, A.: Erratum to: Multivariate McCormick relaxations. J. Global Optim. 68, 219–225 (2017). https://doi.org/10.1007/s10898-016-0470-0

    Article  MathSciNet  MATH  Google Scholar 

  41. Najman, J., Mitsos, A.: Convergence analysis of multivariate McCormick relaxations. J. Global Optim. 66(4), 1–32 (2016). https://doi.org/10.1007/s10898-016-0408-6

    Article  MathSciNet  MATH  Google Scholar 

  42. Najman, J., Mitsos, A.: Convergence order of McCormick relaxations of LMTD function in heat exchanger networks. Comput. Aided Chem. Eng. 38, 1605–1610 (2016). https://doi.org/10.1016/B978-0-444-63428-3.50272-1

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  44. Najman, J., Mitsos, A.: Tighter McCormick relaxations through subgradient propagation. J. Global Optim. 75, 565–593 (2019). https://doi.org/10.1007/s10898-019-00791-0

    Article  MathSciNet  MATH  Google Scholar 

  45. Nedialkov, N.S., Kreinovich, V., Starks, S.A.: Interval arithmetic, affine arithmetic, Taylor series methods: why, what next? Numer. Algorithms 37(1), 325–336 (2004). https://doi.org/10.1023/B:NUMA.0000049478.42605.cf

    Article  MathSciNet  MATH  Google Scholar 

  46. Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR 13(3), 247–277 (2015). https://doi.org/10.1007/s10288-014-0269-0

    Article  MathSciNet  MATH  Google Scholar 

  47. Nohra, C.J., Raghunathan, A.U., Sahinidis, N.: Spectral relaxations and branching strategies for global optimization of mixed-integer quadratic programs. SIAM J. Optim. 31(1), 142–171 (2021). https://doi.org/10.1137/19M1271762

    Article  MathSciNet  MATH  Google Scholar 

  48. Nohra, C.J., Raghunathan, A.U., Sahinidis, N.V.: SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01680-9

    Article  MATH  Google Scholar 

  49. Pakseresht, A.H., Ghasali, E., Nejati, M., Shirvanimoghaddam, K., Javadi, A.H., Teimouri, R.: Development empirical-intelligent relationship between plasma spray parameters and coating performance of Yttria–Stabilized zirconia. Int. J. Adv. Manuf. Technol. 76(5), 1031–1045 (2015). https://doi.org/10.1007/s00170-014-6212-x

    Article  Google Scholar 

  50. Puranik, Y., Sahinidis, N.V.: Domain reduction techniques for global NLP and MINLP optimization. Constraints 22(3), 338–376 (2017). https://doi.org/10.1007/s10601-016-9267-5

    Article  MathSciNet  MATH  Google Scholar 

  51. Rajakumar, S., Balasubramanian, V.: Diffusion bonding of titanium and aa 7075 aluminum alloy dissimilar joints-process modeling and optimization using desirability approach. Int. J. Adv. Manuf. Technol. 86(1), 1095–1112 (2016). https://doi.org/10.1007/s00170-015-8223-7

    Article  Google Scholar 

  52. Rajesh, P., Nagaraju, U., Gowd, G.H., Vardhan, T.V.: Experimental and parametric studies of ND: Yag laser drilling on austenitic stainless steel. Int. J. Adv. Manuf. Technol. 93(1), 65–71 (2017). https://doi.org/10.1007/s00170-015-7639-4

    Article  Google Scholar 

  53. Rump, S.M., Kashiwagi, M.: Implementation and improvements of affine arithmetic. Nonlinear Theory Appl. IEICE 6(3), 341–359 (2015). https://doi.org/10.1587/nolta.6.341

    Article  Google Scholar 

  54. Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996). https://doi.org/10.1007/BF00138693

    Article  MathSciNet  MATH  Google Scholar 

  55. Sahinidis, N.V.: BARON 21.1.13: Global Optimization of Mixed-Integer Nonlinear Programs. User’s Manual (2017). https://www.minlp.com/downloads/docs/baron%20manual.pdf

  56. Sahlodin, A.M., Chachuat, B.: Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs. Appl. Numer. Math. 61(7), 803–820 (2011). https://doi.org/10.1016/j.apnum.2011.01.009

    Article  MathSciNet  MATH  Google Scholar 

  57. Schweidtmann, A.M., Bongartz, D., Grothe, D., Kerkenhoff, T., Lin, X., Najman, J., Mitsos, A.: Deterministic global optimization with gaussian processes embedded. Math. Program. Comput. 13(3), 553–581 (2021). https://doi.org/10.1007/s12532-021-00204-y

    Article  MathSciNet  MATH  Google Scholar 

  58. Schweidtmann, A.M., Bongartz, D., Huster, W.R., Mitsos, A.: Deterministic global process optimization: Flash calculations via artificial neural networks. In: Kiss, A.A., Zondervan, E., Lakerveld, R., Özkan, L. (eds.) Computer Aided Chemical Engineering, vol. 46, pp. 937–942. Elsevier, Amsterdam (2019). https://doi.org/10.1016/b978-0-12-818634-3.50157-0

    Chapter  Google Scholar 

  59. Schweidtmann, A.M., Huster, W.R., Lüthje, J.T., Mitsos, A.: Deterministic global process optimization: accurate (single-species) properties via artificial neural networks. Comput. Chem. Eng. 121, 67–74 (2019). https://doi.org/10.1016/j.compchemeng.2018.10.007

    Article  Google Scholar 

  60. Schweidtmann, A.M., Mitsos, A.: Deterministic global optimization with artificial neural networks embedded. J. Optim. Theory Appl. 180(3), 925–948 (2018). https://doi.org/10.1007/s10957-018-1396-0

    Article  MathSciNet  MATH  Google Scholar 

  61. Scott, J.K., Barton, P.I.: Improved relaxations for the parametric solutions of odes using differential inequalities. J. Global Optim. 57(1), 143–176 (2013). https://doi.org/10.1007/s10898-012-9909-0

    Article  MathSciNet  MATH  Google Scholar 

  62. Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Global Optim. 51(4), 569–606 (2011). https://doi.org/10.1007/s10898-011-9664-7

    Article  MathSciNet  MATH  Google Scholar 

  63. Singer, A.B.: Global dynamic optimization. Ph.D. thesis, Massachusetts Institute of Technology (2004)

  64. Stolfi, J., Figueiredo, L.H.D.: An introduction to affine arithmetic. TEMA Tendências em Matemática Aplicada e Computacional 4(3), 297–312 (2003). https://doi.org/10.5540/tema.2003.04.03.0297

    Article  MathSciNet  MATH  Google Scholar 

  65. Stuber, M.D., Scott, J.K., Barton, P.I.: Convex and concave relaxations of implicit functions. Optim. Methods Softw. 30(3), 424–460 (2015). https://doi.org/10.1080/10556788.2014.924514

    Article  MathSciNet  MATH  Google Scholar 

  66. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005). https://doi.org/10.1007/s10107-005-0581-8

    Article  MathSciNet  MATH  Google Scholar 

  67. Tsoukalas, A., Mitsos, A.: Multivariate McCormick relaxations. J. Global Optim. 59(2–3), 633–662 (2014). https://doi.org/10.1007/s10898-014-0176-0

    Article  MathSciNet  MATH  Google Scholar 

  68. Wang, C., Wilhelm, M.E., Stuber, M.D.: Semi-Infinite optimization with hybrid models. Ind. Eng. Chem. Res. 61(15), 5239–5254 (2022). https://doi.org/10.1021/acs.iecr.2c00113

    Article  Google Scholar 

  69. Wang, E., Zhang, Q., Shen, B., Zhang, G., Lu, X., Wu, Q., Wang, Y.: Intel Math Kernel Library, p. 167. Springer, New York (2014). https://doi.org/10.1007/978-3-319-06486-4_7

    Book  Google Scholar 

  70. Watson, H.A., Vikse, M., Gundersen, T., Barton, P.I.: Optimization of single mixed-refrigerant natural gas liquefaction processes described by nondifferentiable models. Energy 150, 860–876 (2018). https://doi.org/10.1016/j.energy.2018.03.013

    Article  Google Scholar 

  71. Wechsung, A., Scott, J.K., Watson, H.A.J., Barton, P.I.: Reverse propagation of McCormick relaxations. J. Global Optim. 63(1), 1–36 (2015). https://doi.org/10.1007/s10898-015-0303-6

    Article  MathSciNet  MATH  Google Scholar 

  72. Wilhelm, M.E., Gottlieb, R.X., Stuber, M.D.: PSORLab/McCormick.jl (2020). https://doi.org/10.5281/ZENODO.4278415. https://github.com/PSORLab/McCormick.jl

  73. Wilhelm, M.E., Le, A.V., Stuber, M.D.: Global optimization of stiff dynamical systems. AIChE J. 65, 16836 (2019). https://doi.org/10.1002/aic.16836

    Article  Google Scholar 

  74. Wilhelm, M.E., Stuber, M.D.: EAGO.jl easy advanced global optimization in Julia. Optim. Methods Softw. 1, 23 (2020). https://doi.org/10.1080/10556788.2020.1786566

    Article  MATH  Google Scholar 

  75. Wilhelm, M.E., Wang, C., Stuber, M.D.: Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization. J. Glob. Optim. (2022). https://doi.org/10.1007/s10898-022-01228-x

    Article  MATH  Google Scholar 

  76. Yue, Z., Huang, C., Zhu, H., Wang, J., Yao, P., Liu, Z.: Optimization of machining parameters in the abrasive waterjet turning of alumina ceramic based on the response surface methodology. Int. J. Adv. Manuf. Technol. 71(9–12), 2107–2114 (2014). https://doi.org/10.1007/s00170-014-5624-y

    Article  Google Scholar 

  77. Zorn, K., Sahinidis, N.V.: Global optimization of general non-convex problems with intermediate bilinear substructures. Optim. Methods Softw. 29(3), 442–462 (2014). https://doi.org/10.1080/10556788.2013.783032

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 1932723. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We also gratefully acknowledge the Air Force Research Laboratory, Materials and Manufacturing Directorate (AFRL/RXMS) for support via Contract No. FA8650-20-C-5206. The views, opinions, and/or findings contained in this paper are those of the authors and should not be interpreted as representing the official views or policies, either expressed or implied, of the Air Force Research Laboratory, the United States Air Force, or the Department of Defense.

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Wilhelm, M.E., Stuber, M.D. Improved Convex and Concave Relaxations of Composite Bilinear Forms. J Optim Theory Appl 197, 174–204 (2023). https://doi.org/10.1007/s10957-023-02196-2

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