Journal of Global Optimization

, Volume 24, Issue 1, pp 1–33

A Rigorous Global Optimization Algorithm for Problems with Ordinary Differential Equations



The optimization of systems which are described by ordinary differential equations (ODEs) is often complicated by the presence of nonconvexities. A deterministic spatial branch and bound global optimization algorithm is presented in this paper for systems with ODEs in the constraints. Upper bounds for the global optimum are produced using the sequential approach for the solution of the dynamic optimization problem. The required convex relaxation of the algebraic functions is carried out using well-known global optimization techniques. A convex relaxation of the time dependent information is obtained using the concept of differential inequalities in order to construct bounds on the space of solutions of parameter dependent ODEs as well as on their second-order sensitivities. This information is then incorporated in the convex lower bounding NLP problem. The global optimization algorithm is illustrated by applying it to four case studies. These include parameter estimation problems and simple optimal control problems. The application of different underestimation schemes and branching strategies is discussed.

Global optimization Ordinary differential equations Convex underestimation Differential inequalities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, E.: 1980, On methods for the construction of the boundaries of sets of solutions for differential equations or finite-dimensional approximations with input sets. In: G. Alefeld and R. D. Grigorieff (eds.): Fundamentals of numerical computation, Computing Supplement 2. Berlin: Springer, pp. 1–16.Google Scholar
  2. Adjiman, C. S., I. P. Androulakis, and C. A. Floudas: 1998a, A global optimization method, αBB, for general twice-differentiable constrained NLPs¶ II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179.Google Scholar
  3. Adjiman, C. S., S. Dallwig, C. A. Floudas, and A. Neumaier: 1998b, A global optimization method, αBB, for general twice-differentiable constrained NLPs¶ I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158.Google Scholar
  4. Adjiman, C. S. and C. A. Floudas: 1996, Rigorous convex underestimators for general twicedifferentiable problems. J. Global Opt. 9, 23–40.Google Scholar
  5. Allen, M. P. and D. J. Tildesley: 1987, Computer simulation of liquids. Oxford: Clarendon Press.Google Scholar
  6. Androulakis, I. P., C. D. Maranas, and C. A. Floudas: 1995, αBB: A global optimization method for general constrained nonconvex problems. J. Global Opt. 7, 337–363.Google Scholar
  7. Banga, J. R. and W. D. Seider: 1996, Global optimization of chemical processes using stochastic algorithms. In: State of the art in global optimization, Series in nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers, pp. 563–583.Google Scholar
  8. Banks, R. B.: 1994, Growth and diffusion phenomena. Mathematical frameworks and applications. Springer, Berlin.Google Scholar
  9. Berz, M., K. Makino, and J. Hoefkens: 2001, Verified integration of dynamics in the solar system. Nonlinear Analysis: Theory, Methods and Applications 47, 179–190.Google Scholar
  10. Biegler, L. T.: 1984, Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8(3/4), 243–248.Google Scholar
  11. Boender, C. G. E. and H. E. Romeijn: 1995, Stochastic methods. In: R. Horst and P. M. Pardalos (eds.): Handbook of global optimization, Series in nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers, pp. 829–869.Google Scholar
  12. Bojkov, B. and R. Luus: 1993, Evaluation of the parameters used in iterative dynamic programming. Can. J. Chem. Eng. 71, 451–459.Google Scholar
  13. Carrasco, E. F. and J. R. Banga: 1997, Dynamic optimization of batch reactors using adaptive stochastic algorithms. Ind. Eng. Chem. Res. 36(6), 2252–2261.Google Scholar
  14. Coleman, T., M. A. Branch, and A. Grace: 1999, Optimization toolbox. For use with MATLAB. User's guide, Ver. 2. The MathWorks, Inc.Google Scholar
  15. Corliss, G. F.: 1995, Guaranteed error bounds for ordinary differential equations. In: J. Levesley, W. A. Light, and M. Marletta (eds.): Theory of numerics in ordinary and partial differential equations. Oxford: Oxford Univ. Press, pp. 1–75.Google Scholar
  16. Cuthrell, J. E. and L. T. Biegler: 1987, On the optimisation of differential-algebraic process systems. AIChE J. 33(8), 1257–1270.Google Scholar
  17. Dadebo, S. A. and K. B. McAuley: 1995, Dynamic optimization of constrained chemical engineering problems using dynamic programming. Comput. Chem. Eng. 19(5), 513–525.Google Scholar
  18. Esposito, W. and C. Floudas: 2000a, Deterministic global optimization in nonlinear optimal control problems. J. Global Opt. 17(1/4), 97–126.Google Scholar
  19. Esposito, W. R. and C. A. Floudas: 2000b, Global optimization for the parameter estimation of differential-algebraic systems. Ind. Eng. Chem. Res. 39, 1291–1310.Google Scholar
  20. Esposito, W. R. and C. A. Floudas: 2000c, Global optimization of nonconvex problems with differential-algebraic constraints. In: S. Pierucci (ed.): European Symposium on Computer Aided Process Engineering ¶10. pp. 73–78, Elsevier.Google Scholar
  21. Floudas, C. A.: 1999, Deterministic global optimization: Theory, methods and applications, Series in nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers.Google Scholar
  22. Floudas, C. A., P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gümüs, S. T. Harding, J. L. Klepeis, C. A. Meyer, and C. A. Schweiger: 1999, Handbook of test problems in local and global optimization, Series in nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers.Google Scholar
  23. Goh, C. J. and K. L. Teo: 1988, Control parameterization: a unified approach to optimal control problems with general constraints. Automatica 24(1), 3–18.Google Scholar
  24. Horst, R. and H. Tuy: 1996, Global Optimization. Deterministic approaches. Berlin: Springer-Verlag, 3rd edition.Google Scholar
  25. Jackson, L.: 1975, Interval arithmetic error-bounding algorithms. SIAM J. Numer. Anal. 12(2), 223–238.Google Scholar
  26. Krückeberg, F.: 1969, Ordinary differential equations. In: E. Hansen (ed.): Topics in interval analysis. Oxford: Clarendon Press, pp. 91–97.Google Scholar
  27. Kühn, W.: 1998, Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61, 47–67.Google Scholar
  28. Lakshmikantham, V. and S. Leela: 1969, Differential and integral inequalities. Theory and applications, Vol. 55-I of Series in mathematics in science and engineering. New York: Academic Press.Google Scholar
  29. Logsdon, J. S. and L. T. Biegler: 1989, Accurate solution of differential algebraic optimization problems. Ind. Eng. Chem. Res. 28(11), 1628–1639.Google Scholar
  30. Lohner, R. J.: 1987, Enclosing the solutions of ordinary initial and boundary problems. In: E. W. Kaucher, U. W. Kulisch, and C. Ullrich (eds.): Computer arithmetic: Scientific computation and programming languages, Series in Computer Science. Stuttgart: Wiley-Teubner, pp. 255–286.Google Scholar
  31. Luus, R.: 1990a, Application of dynamic programming to high-dimensional nonlinear optimal control problems. Int. J. Control 52(1), 239–250.Google Scholar
  32. Luus, R.: 1990b, Optimal control by dynamic programming using systematic reduction in grid size. Int. J. Control 51(5), 995–1013.Google Scholar
  33. Luus, R.: 1993, Piecewise linear continuous optimal control by iterative dynamic programming. Ind. Eng. Chem. Res. 32, 859–865.Google Scholar
  34. Luus, R. and B. Bojkov: 1994, Global optimization of the bifunctional catalyst problem. Can. J. Chem. Eng. 72, 160–163.Google Scholar
  35. Luus, R. and D. E. Cormack: 1972, Multiplicity of solutions resulting from the use of variational methods in optimal control problems. Can. J. Chem. Eng. 50, 309–311.Google Scholar
  36. Luus, R. and O. Rosen: 1991, Application of iterative dynamic programming to final state constrained optimal control problems. Ind. Eng. Chem. Res. 30(7), 1525–1530.Google Scholar
  37. Luus, R., J. Dittrich, and F. J. Keil: 1992, Multiplicity of solutions in the optimization of a bifunctional catalyst blend in a tubular reactor. Can. J. Chem. Eng. 70, 780–785.Google Scholar
  38. Maranas, C. D. and C. A. Floudas: 1994, Global minimum potential energy conformations of small molecules. J. Global Opt. 4, 135–170.Google Scholar
  39. McCormick, G. P.: 1976, Computability of global solutions to factorable nonconvex programs: Part I·Convex underestimating problems. Mathematical Programming 10, 147–175.Google Scholar
  40. Moore, R. E.: 1966, Interval analysis. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  41. Moore, R. E.: 1978, Bounding sets in function spaces with applications to nonlinear operator equations. SIAM Review 20(3), 492–512.Google Scholar
  42. Moore, R. E.: 1979, Methods and applications of interval analysis. Philadelphia, PA: SIAM.Google Scholar
  43. Moore, R. E.: 1984, A survey of interval methods for differential equations. In: Proceedings of 23rd conference on decision and control (Las Vegas). pp. 1529–1535.Google Scholar
  44. Nedialkov, N. S., K. R. Jackson, and G. F. Corliss: 1999, Validated solutions of initial value problems for ordinary differential equations. Appl. Math. Comput. 105(1), 21–68.Google Scholar
  45. Neumaier, A.: 1993, The wrapping effect, ellipsoid arithmetic, stability and confidence regions. In: R. A. et al. (ed.): Validation Numerics, Vol. 9 of Computing Supplementum. Wien: Springer, pp. 175–190.Google Scholar
  46. Neumaier, A.: 1994, Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory. Computing 52, 315–336.Google Scholar
  47. Nickel, K. L.: 1978, Ein Zusammenhang zwischen Aufgaben monotoner Art und Intervall-Mathematik. In: R. Bulirsch and R. D. Grigorieff (eds.): Numerical treatment of differential equations, Lecture Notes in Computer Science No 631. Berlin: Springer, pp. 121–132.Google Scholar
  48. Nickel, K. L.: 1983, Bounds for the set of solutions of functional-differential equations. Annales Polonici Mathematici 42, 241–257. MRC Technical summary report No 1782, University of Winsconsin, Madison (1977).Google Scholar
  49. Nickel, K. L.: 1985, How to fight the wrapping effect. In: K. L. Nickel (ed.): Interval Mathematics, Lecture Notes in Computer Science No 212. Berlin: Springer, pp. 121–132.Google Scholar
  50. Nickel, K. L.: 1986, Using interval methods for numerical solution of ODE's. ZAMM 66(11), 513–523.Google Scholar
  51. Oh, S. H. and R. Luus: 1977, Use of orthogonal collocation methods in optimal control problems. Int. J. Control 26(5), 657–673.Google Scholar
  52. Pollard, G. P. and R. W. H. Sargent: 1970, Off line computation of optimum controls for a plate distillation column. Automatica 6, 59–76.Google Scholar
  53. Pontryagin, L. S.: 1962, Ordinary differential equations. London: Pergamon Press.Google Scholar
  54. Renfro, J. G., A. M. Morshedi, and O. A. Asbjornsen: 1987, Simultaneous optimization and solution of systems described by differential/algebraic equations. Comput. Chem. Eng. 11(5), 503–517. A RIGOROUS GLOBAL OPTIMIZATION ALGORITHM FOR PROBLEMS WITH ODES 33Google Scholar
  55. Rihm, R.: 1994, On a class of enclosure methods for initial value problems. Computing 53, 369–377.Google Scholar
  56. Rosen, O. and R. Luus: 1992, Global optimization approach to nonlinear optimal control. J. Opt. Th. Appl. 73(3), 547–562.Google Scholar
  57. Rump, S. M.: 1999a, Fast and parallel interval arithmetic. BIT 39(3), 534–554.Google Scholar
  58. Rump, S. M.: 1999b, INTLAB· Interval laboratory. In: T. Csendes (ed.): Developments in reliable computing. Dordrecht: Kluwer Academic Publishers, pp. 77–104.Google Scholar
  59. Sargent, R. W. H.: 2000, Optimal control. J. Comput. Appl. Math. 124(1/2), 361–371.Google Scholar
  60. Sargent, R.W. H. and G. R. Sullivan: 1978, The development of an efficient optimal control package. In: J. Stoer (ed.): Proc. 8th IFIP Conf. Optimisation Tech. Pt. 2. Springer, Berlin: pp. 158–168.Google Scholar
  61. Shampine, L. F. and M. W. Reichelt: 1997, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22.Google Scholar
  62. Smith, E. M. B. and C. C. Pantelides: 1996, Global optimisation of general process models. In: I. E. Grossmann (ed.): Global optimization in engineering design, Series in nonconvex optimization and its applications. Dordrecht: Kluwer Academic Publishers, Ch. 12, pp. 355–386.Google Scholar
  63. Smith, J. M.: 1981, Chemical engineering kinetics. London: McGraw-Hill, 3rd edition.Google Scholar
  64. Stewart, N. F.: 1971, A heuristic to reduce the wrapping effect in the numerical solution of x′ = f (t, x). BIT 11, 328–337.Google Scholar
  65. The MathWorks, Inc.: 1999, Using MATLAB.Google Scholar
  66. Tjoa, I. B. and L. T. Biegler: 1991, Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equation systems. Ind. Eng. Chem. Res. 30, 376–385.Google Scholar
  67. Tsang, T. H., D. M. Himmelblau, and T. F. Edgar: 1975, Optimal control via collocation and nonlinear programming. Int. J. Control 21(5), 763–768.Google Scholar
  68. Vassiliadis, V. S., R. W. H. Sargent, and C. C. Pantelides: 1994a, Solution of a class of multistage dynamic optimization problems. 1. Problems without path constraints. Ind. Eng. Chem. Res. 33(9), 2111–2122.Google Scholar
  69. Vassiliadis, V. S., R. W. H. Sargent, and C. C. Pantelides: 1994b, Solution of a class of multistage dynamic optimization problems. 2. Problems with path constraints. Ind. Eng. Chem. Res. 33(9), 2123–2133.Google Scholar
  70. Villadsen, J. V. and W. E. Stewart: 1967, Solution of boundary-value problems by orthogonal collocation. Chem. Eng. Sci. 22, 1483–1501.Google Scholar
  71. Walter, W.: 1970, Differential and integral inequalities. Berlin: Springer.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  1. 1.Centre for Process Systems Engineering, Department of Chemical Engineering and Chemical TechnologyImperial College of Science, Technology and MedicineLondonUK

Personalised recommendations