Abstract
A deterministic spatial branch and bound global optimization algorithm for problems with ordinary differential equations in the constraints has been developed by Papamichail and Adjiman [A rigorous global optimization algorithm for problems with ordinary differential equations. J. Glob. Optim. 24, 1–33]. In this work, it is shown that the algorithm is guaranteed to converge to the global solution. The proof is based on showing that the selection operation is bound improving and that the bounding operation is consistent. In particular, it is shown that the convex relaxation techniques used in the algorithm for the treatment of the dynamic information ensure bound improvement and consistency are achieved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C.S. Adjiman I.P. Androulakis C.A. Floudas (1998) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs-II Implementation and computational results. Computers and Chemical Engineering 22 IssueID9 1159–1179 Occurrence Handle10.1016/S0098-1354(98)00218-X
C.S. Adjiman S. Dallwig C.A. Floudas A. Neumaier (1998) ArticleTitleA global optimization method, αBB, for general twice-differentiable constrained NLPs-I Theoretical advances Computers and Chemical Engineering 22 IssueID9 1137–1158 Occurrence Handle10.1016/S0098-1354(98)00027-1
C.S. Adjiman C.A. Floudas (1996) ArticleTitleRigorous convex underestimators for general twice-differentiable problems Journal of Global Optimization 9 23–40 Occurrence Handle10.1007/BF00121749
F.A. Al-Khayyal J.E. Falk (1983) ArticleTitleJointly constrained biconvex programming Mathematics of Operations Research 8 IssueID2 273–286
I.P. Androulakis C.D. Maranas C.A. Floudas (1995) ArticleTitleαBB: A global optimization method for general constrained nonconvex problems Journal of Global Optimization 7 337–363 Occurrence Handle10.1007/BF01099647
Barton, P.I. and Lee, C.K. (2003), Global dynamic optimization of linear time varying hybrid systems, Dynamics of Continous Discrete and Impulsive Systems – Series B – Applications and Algorithms pp. 153–158.
L.T. Biegler (1984) ArticleTitleSolution of dynamic optimization problems by successive quadratic programming and orthogonal collocation Computers and Chemical Engineering 8 IssueID3/4 243–248 Occurrence Handle10.1016/0098-1354(84)87012-X
W.R. Esposito C.A. Floudas (2000) ArticleTitleDeterministic global optimization in nonlinear optimal control problems Journal of Global Optimization 17 IssueID1/4 97–126 Occurrence Handle10.1023/A:1026578104213
W.R. Esposito C.A. Floudas (2000) ArticleTitleGlobal optimization for the parameter estimation of differential-algebraic systems Industrial and Engineering Chemistry Research 39 1291–1310 Occurrence Handle10.1021/ie990486w
J.E. Falk R.M. Soland (1969) ArticleTitleAn algorithm for separable nonconvex programming problems Management Science 15 IssueID9 550–569
C.J. Goh K.L. Teo (1988) ArticleTitleControl parameterization: a unified approach to optimal control problems with general constraints Automatica 24 IssueID1 3–18 Occurrence Handle10.1016/0005-1098(88)90003-9
Horst, R. and Tuy, H. (1996), Global Optimization. Deterministic Approaches. Springer-Verlag third edition, Berlin.
C.D. Maranas C.A. 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 problem Mathematical Progamming 10 147–175 Occurrence Handle10.1007/BF01580665
R.E. Moore (1966) Interval Analysis Prentice-Hall Englewood Cliffs, N.J
S.H. Oh R. Luus (1977) ArticleTitleUse of orthogonal collocation methods in optimal control problems International Journal of Control 26 IssueID5 657–673
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
I. Papamichail C.S. Adjiman (2004) ArticleTitleGlobal optimization of dynamic systems Computers and Chemical Engineering 28 IssueID3 403–415 Occurrence Handle10.1016/S0098-1354(03)00195-9
G.P. Pollard R.W.H. Sargent (1970) ArticleTitleOff line computation of optimum controls for a plate distillation column Automatica 6 59–76 Occurrence Handle10.1016/0005-1098(70)90075-0
R.W.H. Sargent (2000) ArticleTitleOptimal control Journal of Computational and Applied Mathematics 124 IssueID1/2 361–371 Occurrence Handle10.1016/S0377-0427(00)00418-0
Sargent, R.W.H. and Sullivan, G.R. (1978), The development of an efficient optimal control package. In: J. Stoer (ed.), Proceeding of 8th IFIP Conference on Optimization Techniques Part 2. Berlin, pp. 158–168, Springer-Verlag.
A.B. Singer P.I. Barton (2004) ArticleTitleGlobal solution of optimization problems with parameter-embedded linear dynamic systems Journal of Optimization Theory and Applications 121 IssueID3 613–646 Occurrence Handle10.1023/B:JOTA.0000037606.79050.a7
Smith, E.M.B. and Pantelides, C.C. (1996), Global optimization of general process models, In: I.E. Grossmann (ed.), Global Optimization in Engineering Design, Series in nonconvex optimization and its applications. Kluwer Academic Publishers, Dordrecht Chapt. 12, pp. 355–386.
T.H. Tsang D.M. Himmelblau T.F. Edgar (1975) ArticleTitleOptimal control via collocation and non-linear programming International Journal of Control 21 IssueID5 763–768
V.S. Vassiliadis R.W.H. Sargent C.C. Pantelides (1994) ArticleTitleSolution of a class of multistage dynamic optimization problems. 1. Problems without path constraints Industrial and Engineering Chemistry Research 33 IssueID9 2111–2122 Occurrence Handle10.1021/ie00033a014
W. Walter (1970) Differential and Integral Inequalities Springer-Verlag Berlin
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Papamichail, I., Adjiman, C.S. Proof of Convergence for a Global Optimization Algorithm for Problems with Ordinary Differential Equations. J Glob Optim 33, 83–107 (2005). https://doi.org/10.1007/s10898-004-6100-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10898-004-6100-2