Abstract
This paper examines global optimization of an integral objective function subject to nonlinear ordinary differential equations. Theory is developed for deriving a convex relaxation for an integral by utilizing the composition result defined by McCormick (Mathematical Programming 10, 147–175, 1976) in conjunction with a technique for constructing convex and concave relaxations for the solution of a system of nonquasimonotone ordinary differential equations defined by Singer and Barton (SIAM Journal on Scientific Computing, Submitted). A fully automated implementation of the theory is briefly discussed, and several literature case study problems are examined illustrating the utility of the branch-and-bound algorithm based on these relaxations.
Similar content being viewed by others
References
M. Adams V. Guillemin (1996) Measure Theory and Probability Birkhäuser Boston
C. Adjiman S. Dallwig C. Floudas (1998a) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs – II. Implementation and computational results Computers and Chemical Engineering 22 IssueID9 1159–1179
C. Adjiman S. Dallwig C. Floudas A. Neumaier (1998b) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs – I. Theoretical advances Computers and Chemical Engineering 22 IssueID9 1137–1158
E. Arthur Bryson SuffixJr. Y.-C. Ho (1975) Applied Optimal Control Taylor & Francis Briston, Pennsylvania
R. Brusch R. Schappelle (1973) ArticleTitleSolution of highly constrained optimal control problems using nonlinear programming AIAA Journal 11 IssueID2 135–136
W.R. Esposito C.A. Floudas (2000a) ArticleTitleDeterministic global optimization in nonlinear optimal control problems Journal of Global Optimization 17 97–126 Occurrence Handle10.1023/A:1026578104213
W.R. Esposito C.A. Floudas (2000b) ArticleTitleGlobal optimization for the parameter estimation of differential-algebraic systems Industrial and Engineering Chemistry Research 39 1291–1310
W. Feehery J. Tolsma P. Barton (1997) ArticleTitleEfficient sensitivity analysis of large-scale differential-algebraic systems Applied Numerical Mathematics 25 IssueID1 41–54 Occurrence Handle10.1016/S0168-9274(97)00050-0
C.A. Floudas P.M. Pardalos C.S. Adjiman W.R. Esposito Z.H. Gumus S.T. Harding J.L. Klepeis C.A. Meyer C.A. Schweiger (1999) Handbook of Test Problems in Local and Global Optimization Kluwer Academic Publishers Dordrecht
Gill, P.E., Murray, W., Saunders, M.A., and Wright, M.H. (1998), User’s Guide for NPSOL 5.0: A Fortran Package for Nonlinear Programming. Technical report, Stanford University.
Harrison, G. (1977), Dynamic models with uncertain parameters, In: Avula, X. (ed.), Proceedings of the First International Conference on Mathematical Modeling, vol. 1, pp. 295–304.
Hindmarsh, A.C. and Serban, R. (2002), User Documentation for CVODES, An ODE Solver with Sensitivity Analysis Capabilities. Technical report, Lawrence Livermore National Laboratory.
R. Horst H. Tuy (1993) Global Optimization Springer-Verlag Berlin
R. Luus (1990) ArticleTitleOptimal control by dynamic programming using systematic reduction in grid size International Journal of Control 5 995–1013
C. Maranas C. Floudas (1994) ArticleTitleGlobal minimum potential energy conformations of small molecules Journal of Global Optimization 4 135–170 Occurrence Handle10.1007/BF01096720
G.P. McCormick (1976) ArticleTitleComputability of global solutions to factorable nonconvex programs: Part I – convex underestimating problems Mathematical Programming 10 147–175 Occurrence Handle10.1007/BF01580665
C. Neuman A. Sen (1973) ArticleTitleA suboptimal control algorithm for constrained problems using cubic splines Automatica 9 601–613 Occurrence Handle10.1016/0005-1098(73)90045-9
I. Papamichail C.S. Adjiman (2002) ArticleTitleA rigorous global optimization algorithm for problems with ordinary differential equations Journal of Global Optimization 24 1–33 Occurrence Handle10.1023/A:1016259507911
L. Pontriagin (1962) The Mathematical Theory of Optimal Processes Interscience Publishers New York
H. Ryoo N. Sahinidis (1995) ArticleTitleGlobal optimization of nonconvex NLPs and MINLPs with application in process design Computers and Chemical Engineering 19 IssueID5 551–566 Occurrence Handle10.1016/0098-1354(94)00097-8
Singer, A.B. (2004a). Global Dynamic Optimization, Ph.D. thesis, Massachusetts Institute of Technology.
Singer, A.B. (2004b). LibBandB Version 3.2 Manual, Technical report, Massachusetts Institute of Technology.
A.B. Singer P.I. Barton (2004) ArticleTitleGlobal Solution of linear dynamic embedded optimization problems Journal of Optimization Theory and Applications 121 IssueID3 613–646 Occurrence Handle10.1023/B:JOTA.0000037606.79050.a7
K. Teo G. Goh K. Wong (1991) A Unified Computational Approach to Optimal Control Problems. Pitman Monographs and Surveys in Pure and Applied Mathematics John Wiley & Sons New York
I. Tjoa L. Biegler (1991) ArticleTitleSimultaneous solution and optimization strategies for parameter-estimation of differential-algebraic equation systems Industrial and Engineering Chemistry Research 30 IssueID2 376–385 Occurrence Handle10.1021/ie00050a015
J.L. Troutman (1996) Variational Calculus and Optimal Control: Optimization with Elementary Convexity EditionNumber2 Springer-Verlag New York
T. Tsang D. Himmelblau T. Edgar (1975) ArticleTitleOptimal control via collocation and nonlinear programming International Journal of Control 21 763–768
W. Walter (1970) Differential and Integral Inequalities Springer-Verlag Berlin
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Singer, A.B., Barton, P.I. Global Optimization with Nonlinear Ordinary Differential Equations. J Glob Optim 34, 159–190 (2006). https://doi.org/10.1007/s10898-005-7074-4
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-005-7074-4