A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations


Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart–Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.


  1. 1.

    Aspen Technology, Inc (2009) Aspen Simulation Workbook, Version Number: V7.1. Burlington, MA, USA. EO and SM Variables and Synchronization, p. 110

  2. 2.

    Auger, A., Hansen, N.: A restart CMA evolution strategy with increasing population size. In: 2005. The 2005 IEEE Congress on Evolutionary Computation, vol. 2, pp 1769–1776. IEEE (2005)

  3. 3.

    Bachmann, B., Aronßon, P, Fritzson, P.: Robust initialization of differential algebraic equations. In: 1st International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools, vol. 2007, pp 151–163, Linköping University Electronic Press; Linköpings universitet, Linköping Electronic Conference Proceedings (2007)

  4. 4.

    Baharev, A.: https://sdopt-tearing.readthedocs.io, Exact and heuristic methods for tearing (2016)

  5. 5.

    Baharev, A., Neumaier, A.: A globally convergent method for finding all steady-state solutions of distillation columns. AIChE J. 60, 410–414 (2014)

    Article  Google Scholar 

  6. 6.

    Baharev, A., Kolev, L., Rév, E: Computing multiple steady states in homogeneous azeotropic and ideal two-product distillation. AIChE J. 57, 1485–1495 (2011)

    Article  Google Scholar 

  7. 7.

    Baharev, A., Domes, F., Neumaier, A.: Online supplementary material of the present manuscript. http://www.baharev.info/finding_all_solutions.html (2016a)

  8. 8.

    Baharev, A., Schichl, H., Neumaier, A.: Decomposition methods for solving nonlinear systems of equations, http://reliablecomputing.eu/baharev_tearing_survey.pdf, submitted (2016b)

  9. 9.

    Baharev, A., Schichl, H., Neumaier, A.: Ordering matrices to bordered lower triangular form with minimal border width, http://reliablecomputing.eu/baharev_tearing_exact_algorithm.pdf, submitted (2016c)

  10. 10.

    Bates, D.J., Hauenstein, J.D., Sommese, A.J.: Efficient path tracking methods. Numer. Algorithm. 58(4), 451–459 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically Solving Polynomial Systems with Bertini, Software, Environments and Tools, vol 25. SIAM, Philadelphia, PA (2013)

  12. 12.

    Bates, D.J., Newell, A.J., Niemerg, M.: BertiniLab: A MATLAB interface for solving systems of polynomial equations. Numer. Algorithm. 71(1), 229–244 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Beelitz, T., Frommer, A., Lang, B., Willems, P.: Symbolic–numeric techniques for solving nonlinear systems. PAMM 5(1), 705–708 (2005)

    Article  Google Scholar 

  14. 14.

    Bekiaris, N., Meski, G.A., Radu, C.M., Morari, M.: Multiple steady states in homogeneous azeotropic distillation. Ind. Eng. Chem. Res. 32, 2023–2038 (1993)

    Article  Google Scholar 

  15. 15.

    Boston, J.F., Sullivan, S.L.: A new class of solution methods for multicomponent, multistage separation processes. Can. J. Chem. Eng. 52, 52–63 (1974)

    Article  Google Scholar 

  16. 16.

    Christensen, J.H.: The structuring of process optimization. AIChE J. 16(2), 177–184 (1970)

    Article  Google Scholar 

  17. 17.

    Dassault Systèmes, AB: Dymola—Dynamic Modeling Laboratory. User Manual. Vol. 2., Ch. 8. Advanced Modelica Support (2014)

  18. 18.

    Davis, T.A.: Direct methods for sparse linear systems. In: Higham, N.J. (ed.) Fundamentals of Algorithms. SIAM, Philadelphia, USA (2006)

  19. 19.

    Dietmaier, P.: The Stewart-Gough platform of general geometry can have 40 real postures, pp 7–16. Springer, Netherlands, Dordrecht (1998)

  20. 20.

    Doedel, E.J., Wang, X.J., Fairgrieve, T.F.: AUTO94: Software for continuation and bifurcation problems in ordinary differential equations. Technical Report CRPC-95-1, Center for Research on Parallel Computing, California Institute of Technology, Pasadena CA 91125 (1995)

  21. 21.

    Doherty, M.F., Fidkowski, Z.T., Malone, M.F., Taylor, R.: Perry’s Chemical Engineers’ Handbook, 8th edn., p 33. McGraw-Hill Professional (2008). chapter 13

  22. 22.

    Dorigo, M., Birattari, M., Stützle, T: Ant colony optimization. IEEE Comput. Intell. Mag. 1(4), 28–39 (2006)

    Article  Google Scholar 

  23. 23.

    Dorn, C., Güttinger, T E, Wells, G.J., Morari, M.: Stabilization of an unstable distillation column. Ind. Eng. Chem. Res. 37, 506–515 (1998)

    Article  Google Scholar 

  24. 24.

    Duff, I.S., Erisman, A.M., Reid, J.K.: Direct methods for sparse matrices. Clarendon Press, Oxford (1986)

  25. 25.

    Dulmage, A.L., Mendelsohn, N.S.: Coverings of bipartite graphs. Can. J. Math. 10, 517–534 (1958)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Dulmage, A.L., Mendelsohn, N.S.: A structure theory of bipartite graphs of finite exterior dimension. Trans. Royal Soc. Can. Sec. 3(53), 1–13 (1959)

    MATH  Google Scholar 

  27. 27.

    Dulmage, A.L., Mendelsohn, N.S.: Two Algorithms for Bipartite Graphs. J. Soc. Ind. Appl. Math. 11, 183–194 (1963)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995. MHS’95, pp 39–43. IEEE (2002)

  29. 29.

    Erisman, A.M., Grimes, R.G., Lewis, J.G., Poole, W.G.J.: A structurally stable modification of Hellerman-Rarick’s P 4 algorithm for reordering unsymmetric sparse matrices. SIAM J. Numer. Anal. 22, 369–385 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Faugère, J C, Lazard, D.: Combinatorial classes of parallel manipulators. Mech Mach. Theory 30(6), 765–776 (1995)

    Article  Google Scholar 

  31. 31.

    Fletcher, R., Hall, J.A.J.: Ordering algorithms for irreducible sparse linear systems. Ann. Oper. Res. 43, 15–32 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Fourer, R.: Staircase matrices and systems. SIAM Rev. 26(1), 1–70 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, Brooks/Cole, USA (2003)

  34. 34.

    Fritzson, P.: Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press (2004)

  35. 35.

    Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore, USA (1996)

  36. 36.

    gPROMS. Process Systems Enterprise Limited, gPROMS. http://www.psenterprise.com, [Online; accessed 17-November-2015] (2015)

  37. 37.

    Gupta, P.K., Westerberg, A.W., Hendry, J.E., Hughes, R.R.: Assigning output variables to equations using linear programming. AIChE J.ournal 20(2), 397–399 (1974)

    Article  Google Scholar 

  38. 38.

    Güttinger, T E, Morari, M.: Comments on multiple steady states in homogeneous azeotropic distillation. Ind. Eng. Chem. Res. 35, 2816–2816 (1996)

    Article  Google Scholar 

  39. 39.

    Güttinger, T E, Dorn, C., Morari, M.: Experimental study of multiple steady states in homogeneous azeotropic distillation. Ind. Eng. Chem. Res. 36, 794–802 (1997)

    Article  Google Scholar 

  40. 40.

    Guzman, Y.A., Hasan, M.M.F., Floudas, C.A.: Computational comparison of convex underestimators for use in a branch-and-bound global optimization framework. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp 229–246. Springer, New York, USA (2014)

  41. 41.

    Gwaltney, C.R., Lin, Y., Simoni, L.D., Stadtherr, M.A.: Interval methods for nonlinear equation solving applications. Wiley, Chichester, UK (2008)

  42. 42.

    Hellerman, E., Rarick, D.C.: Reinversion with preassigned pivot procedure. Math Programm. 1, 195–216 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Hellerman, E., Rarick, D.C.: The partitioned preassigned pivot procedure (P 4). In: Rose, D.J., Willoughby, R.A. (eds.) Sparse Matrices and their Applications, The IBM Research Symposia Series, pp 67–76. Springer, US (1972)

  44. 44.

    HSL: A collection of Fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk (2016)

  45. 45.

    Johnson, D.M., Dulmage, A.L., Mendelsohn, N.S.: Connectivity and reducibility of graphs. Can. J. Math. 14, 529–539 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Kannan, A., Joshi, M.R., Reddy, G.R., Shah, D.M.: Multiple-steady-states identification in homogeneous azeotropic distillation using a process simulator. Ind. Eng. Chem. Res. 44, 4386–4399 (2005)

    Article  Google Scholar 

  47. 47.

    Kearfott, R.B.: Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems. Computing 47(2), 169–191 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Kröner, A, Marquardt, W., Gilles, E.: Getting around consistent initialization of DAE systems? Comput. Chem. Eng. 21(2), 145–158 (1997)

    Article  Google Scholar 

  49. 49.

    Lazard, D.: On the representation of rigid-body motions and its application to generalized platform manipulators, pp 175–181. Springer, Netherlands, Dordrecht (1993)

  50. 50.

    Lewis, W.K., Matheson, G.L.: Studies in distillation. Ind. Eng. Chem. 24, 494–498 (1932)

    Article  Google Scholar 

  51. 51.

    Malinen, I., Tanskanen, J.: Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies. Comput. Chem. Eng. 34(11), 1761–1774 (2010)

    Article  Google Scholar 

  52. 52.

    Mattsson, S., Elmqvist, H., Otter, M.: Physical system modeling with Modelica. Control Eng. Pract. 6, 501—-510 (1998)

    Article  Google Scholar 

  53. 53.

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

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Modelica: Modelica and the modelica association. https://www.modelica.org/, [Online; accessed 10-October-2016] (2016)

  55. 55.

    Modelon, A.B.: JModelica.org User Guide, verison 1.17. http://www.jmodelica.org/page/236, [Online; accessed 10-October-2016] (2016)

  56. 56.

    Mourrain, B.: The 40 g̈eneric positions of a parallel robot. In: Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, ACM, NY, USA, ISSAC ’93, pp. 173–182, doi:10.1145/164081.164120 (1993)

  57. 57.

    Naphthali, L.M., Sandholm, D.P.: Multicomponent separation calculations by linearization. AIChE J. 17, 148–153 (1971)

    Article  Google Scholar 

  58. 58.

    Neumaier, A.: Interval methods for systems of equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  59. 59.

    Neumaier, A., Azmi, B.: LMBOPT – A limited memory method for bound-constrained optimization, http://www.mat.univie.ac.at/neum/ms/lmbopt.pdf, in preparation (2017)

  60. 60.

    Ochel, L.A., Bachmann, B.: Initialization of equation-based hybrid models within OpenModelica. In: 5th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools (University of Nottingham), pp 97–103. Linköping University Electronic Press; Linköpings universitet, Linköping Electronic Conference Proceedings, Nottingham, Uk (2013)

  61. 61.

    OpenModelica: Openmodelica user’s guide. https://openmodelica.org/doc/OpenModelicaUsersGuide/latest/omchelptext.html, [Online; accessed 10-October-2016] (2016)

  62. 62.

    De P Soares, R., Secchi, A.R.: EMSO: A new environment for modelling, simulation and optimisation. In: Computer Aided Chemical Engineering, vol. 14, pp 947–952. Elsevier (2003)

  63. 63.

    Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9(2), 213–231 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

    Petlyuk, F.B.: Distillation theory and its application to optimal design of separation units. Cambridge University Press, Cambridge, UK (2004)

  65. 65.

    Piela, P.C., Epperly, T.G., Westerberg, K.M., Westerberg, A.W.: ASCEND: An object-oriented computer environment for modeling and analysis: the modeling language. Comput. Chem. Eng. 15(1), 53–72 (1991)

    Article  Google Scholar 

  66. 66.

    Pothen, A., Fan, C.J.: Computing the block triangular form of a sparse matrix. ACM Trans. Math. Softw. 16, 303–324 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  67. 67.

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

    MathSciNet  MATH  Article  Google Scholar 

  68. 68.

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

    MathSciNet  MATH  Article  Google Scholar 

  69. 69.

    Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science. http://www.mat.univie.ac.at/neum/glopt/coconut/Benchmark/Benchmark.html, vol. 2861, pp 211–222. Springer, Berlin Heidelberg (2003)

  70. 70.

    Sielemann, M.: Device-oriented modeling and simulation in aircraft energy systems design. Dissertation, TU Hamburg, Hamburg (2012). 10.15480/882.1111

  71. 71.

    Sielemann, M., Schmitz, G.: A quantitative metric for robustness of nonlinear algebraic equation solvers. Math. Comput. Simul. 81(12), 2673–2687 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    Sielemann, M., Casella, F., Otter, M.: Robustness of declarative modeling languages: improvements via probability-one homotopy. Simul. Modell. Pract. Theory 38, 38–57 (2013)

    Article  Google Scholar 

  73. 73.

    Smith, L.: Improved placement of local solver launch points for large-scale global optimization. PhD thesis, Ottawa-Carleton Institute for Electrical and Computer Engineering (OCIECE). Carleton University, Ontario, Canada (2011)

  74. 74.

    Smith, L., Chinneck, J., Aitken, V.: Constraint consensus concentration for identifying disjoint feasible regions in nonlinear programmes. Optim. Methods Softw. 28(2), 339–363 (2013a)

    MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    Smith, L., Chinneck, J., Aitken, V.: Improved constraint consensus methods for seeking feasibility in nonlinear programs. Comput. Optim. Appl. 54(3), 555–578 (2013b)

    MathSciNet  MATH  Article  Google Scholar 

  76. 76.

    Soares, R.P.: Finding all real solutions of nonlinear systems of equations with discontinuities by a modified affine arithmetic. Comput. Chem. Eng. 48, 48–57 (2013)

    Article  Google Scholar 

  77. 77.

    Sommese, A.J., Wampler II, C.W.: The numerical solution of systems of polynomials arising in engineering and science. World Scientific (2005)

  78. 78.

    Stadtherr, M.A., Wood, E.S.: Sparse matrix methods for equation-based chemical process flowsheeting–I: Reordering phase. Comput. Chem. Eng. 8(1), 9–18 (1984a)

    Article  Google Scholar 

  79. 79.

    Stadtherr, M.A., Wood, E.S.: Sparse matrix methods for equation-based chemical process flowsheeting–II: Numerical Phase. Comput. Chem. Eng. 8(1), 19–33 (1984b)

    Article  Google Scholar 

  80. 80.

    Steward, D.V.: Partitioning and tearing systems of equations. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2(2), 345–365 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  81. 81.

    Thiele, E., Geddes, R.: Computation of distillation apparatus for hydrocarbon mixtures. Ind. Eng. Chem. 25, 289–295 (1933)

    Article  Google Scholar 

  82. 82.

    Tiller, M.: Introduction to physical modeling with Modelica. Springer Science & Business Media (2001)

  83. 83.

    Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., Martí, R: Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization. INFORMS J. Comput. 19(3), 328–340 (2007). doi:10.1287/ijoc.1060.0175

  84. 84.

    Unger, J., Kröner, A, Marquardt, W.: Structural analysis of differential-algebraic equation systems —– theory and applications. Comput. Chem. Eng. 19(8), 867–882 (1995)

    Article  Google Scholar 

  85. 85.

    Vadapalli, A., Seader, J.D.: A generalized framework for computing bifurcation diagrams using process simulation programs. Comput. Chem. Eng. 25, 445–464 (2001)

    Article  Google Scholar 

  86. 86.

    Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999)

    MATH  Article  Google Scholar 

  87. 87.

    Verschelde, J.: Polynomial homotopy continuation with phcpack. ACM Commun. Comput. Algebra 44(3/4), 217–220 (2011)

    MATH  Google Scholar 

  88. 88.

    Verschelde, J.: The database of polynomial systems. http://homepages.math.uic.edu/jan/demo.html (2016)

  89. 89.

    Vieira, R. Jr, E. B.: Direct methods for consistent initialization of DAE systems. Comput. Chem. Eng. 25(9–10), 1299–1311 (2001)

  90. 90.

    Vu, X.H., Schichl, H., Sam-Haroud, D.: Interval propagation and search on directed acyclic graphs for numerical constraint solving. J. Glob. Optim. 45(4), 499 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  91. 91.

    Wächter, A, Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Programm. 106, 25–57 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  92. 92.

    Wampler, C.W.: Forward displacement analysis of general six-in-parallel sps (Stewart) platform manipulators using soma coordinates. Mech. Mach. Theory 31 (3), 331–337 (1996)

    Article  Google Scholar 

  93. 93.

    Westerberg, A.W., Edie, F.C.: Computer-aided design, Part 1 enhancing convergence properties by the choice of output variable assignments in the solution of sparse equation sets. Chem. Eng. J. 2, 9–16 (1971a)

    Article  Google Scholar 

  94. 94.

    Westerberg AW, Edie FC: Computer-aided design, part 2 an approach to convergence and tearing in the solution of sparse equation sets. Chem. Eng. J. 2(1), 17–25 (1971b)

  95. 95.

    Wu, W., Reid, G.: Finding points on real solution components and applications to differential polynomial systems. In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, NY, USA, ISSAC ’13, pp 339–346 (2013)

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Baharev, A., Domes, F. & Neumaier, A. A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations. Numer Algor 76, 163–189 (2017). https://doi.org/10.1007/s11075-016-0249-x

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  • Decomposition methods
  • Diakoptics
  • Large-scale systems of equations
  • Numerical instability
  • Sparse matrices
  • Tearing