Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs

Applied Numerical Mathematics

Volumn 61, Issue 7, 2011, Pages 803-820

  Sahlodin, Ali M ^a   Chachuat, Benoît ^a,b  

^a MCMASTER UNIVERSITY   (Canada)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Convex relaxations; Dynamic systems; Global optimization; Interval analysis; McCormick relaxations; Ordinary differential equations

Indexed keywords

CONVEX RELAXATION; DYNAMIC SYSTEMS; INTERVAL ANALYSIS; INTERVAL METHODS; MCCORMICK; NONLINEAR ODES; NUMERICAL EXAMPLE; OBJECT ORIENTED; REFERENCE PARAMETERS; RELAX METHOD; RELAXATION TECHNIQUES; SECOND PHASE;

CONVEX OPTIMIZATION; DYNAMICAL SYSTEMS; GLOBAL OPTIMIZATION; NUMERICAL METHODS; OPTIMIZATION; RELAXATION PROCESSES;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 79953779034     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2011.01.009     Document Type: Article

References (33)

1. C.S. Adjiman, S. Dallwig, C.A. Floudas, and A. Neumaier A global optimization method, αBB, for general twice-differentiable constrained NLPs I. Theoretical advances Comput. Chem. Eng. 22 1998 1137 1158
2. G. Alefeld, and G. Mayer Interval analysis: Theory and applications J. Comput. Appl. Math. 121 2000 421 464
3. M. Berz, and K. Makino Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor series Reliab. Comput. 4 1998 361 369 (Pubitemid 128517412)
4. B. Chachuat, and M.A. Latifi A new approach in deterministic global optimization of problems with ordinary differential equations C.A. Floudas, P.M. Pardalos, Frontiers in Global Optimization Nonconvex Optimization and Its Applications vol. 74 2003 Kluwer Academic Publishers 83 108
5. B. Chachuat, A.B. Singer, and P.I. Barton Global mixed-integer dynamic optimization AIChE J. 51 2005 2235 2253 (Pubitemid 41328519)
6. B. Chachuat, A.B. Singer, and P.I. Barton Global methods for dynamic optimization and mixed-integer dynamic optimization Ind. Eng. Chem. Res. 45 2006 8373 8392 (Pubitemid 46057388)
7. G.F. Corliss, and R. Rihm Validating an a priori enclosure using high-order Taylor series G. Alefeld, A. Frommer, B. Lang, Proc. International Symposium on Scientific Computing Computer Arithmetic and Validated Numerics - SCAN'95 1996 Akademie Verlag Berlin, Germany 228 238 (Pubitemid 126063877)
8. W.R. Esposito, and C.A. Floudas Global optimization for the parameter estimation of differential-algebraic systems Ind. Eng. Chem. Res. 39 2000 1291 1310 (Pubitemid 30265962)
9. G.W. Harrison Dynamic models with uncertain parameters X. Avula, Proc. 1st Intl. Conf. on Mathematical Modeling vol. 1 1977 University of Missouri Rolla 295 304
10. J.B. Hiriart-Urruty, and C. Lemarechal Fundamentals of Convex Analysis 2001 Springer-Verlag Berlin
11. K.R. Jackson, and N.S. Nedialkov Some recent advances in validated methods for IVPs for ODEs Appl. Numer. Math. 42 2002 269 284 (Pubitemid 34782556)
12. R.B. Kearfott Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization Optim. Meth. Soft 21 2006 715 731
13. C.K. Lee, and P.I. Barton Global optimization of linear hybrid systems with varying transition times SIAM J. Control Optim. 47 2008 791 816
14. Y. Lin, and M.A. Stadtherr Validated solutions of initial value problems for parametric ODEs Appl. Numer. Math. 57 2007 1145 1162 (Pubitemid 47126074)
15. R.J. Lohner Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems J.R. Cash, I. Gladwell, Computational Ordinary Differential Equations vol. 1 1992 Clarendon Press 425 436
16. G.P. McCormick Computability of global solutions to factorable nonconvex programs: Part I - Convex underestimating problems Math. Program. 10 1976 147 175
17. A. Mitsos, B. Chachuat, and P.I. Barton McCormick-based relaxations of algorithms SIAM J. Optim. 20 2009 573 601
18. A. Mitsos, B. Chachuat, and P.I. Barton Towards global bilevel dynamic optimization J. Global. Optim. 45 2009 63 93
19. R.E. Moore Methods and Applications of Interval Analysis 1979 SIAM Philadelphia, PA
20. R.E. Moore, R.B. Kearfott, and M.J. Cloud Introduction to Interval Analysis 2009 SIAM Philadelphia, PA
21. N. Nedialkov Interval tools for ODEs and DAEs Proc. 12th GAMM-IMACS Int. Symp. SCAN 2006 2006 IEEE Computer Society
22. N.S. Nedialkov, K.R. Jackson, and G.F. Corliss Validated solution of initial value problems for ordinary differential equations Appl. Math. Comput. 105 1999 21 68 (Pubitemid 129525434)
23. N.S. Nedialkov, K.R. Jackson, and J.D. Pryce An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE Reliab. Comput. 7 2001 449 465 (Pubitemid 34254783)
24. M. Neher, K.R. Jackson, and N.S. Nedialkov On Taylor model based integration of ODEs SIAM J. Numer. Anal. 45 2007 236 262 (Pubitemid 351321966)
25. I. Papamichail, and C.S. Adjiman A rigorous global optimization algorithm for problems with ordinary differential equations J. Global Optim. 24 2002 1 33
26. R. Rihm On a class of enclosure methods for initial value problems Computing 53 1994 369 377
27. A.M. Sahlodin, and B. Chachuat Discretize-then-relax approach for state relaxations in global dynamic optimization S. Pierucci, G. Buzzi Ferraris, Proc 20th European Symposium on Computer Aided Process Engineering - ESCAPE20 Computer Aided Chemical Engineering vol. 28 2010 Elsevier 427 432
28. J.K. Scott, and P.I. Barton Tight, efficient bounds on the solutions of chemical kinetics models Comput. Chem. Eng. 34 2010 717 731
29. J.K. Scott, B. Chachuat, and P.I. Barton Nonlinear convex and concave relaxations for the solutions of parametric ODEs SIAM J. Control Optim. 2010 in revision
30. J.K. Scott, M.D. Stuber, P.I. Barton, Generalized McCormick relaxations for ODEs, J. Global Optim. (2011), in press, doi:10.1007/s10898-011-9664-7.
31. A.B. Singer, and P.I. Barton Bounding the solutions of parameter dependent nonlinear ordinary differential equations SIAM J. Sci. Comput. 27 2006 2167 2182 (Pubitemid 44784233)
32. A.B. Singer, and P.I. Barton Global optimization with nonlinear ordinary differential equations J. Global Optim. 34 2006 159 190 (Pubitemid 43269381)
33. E.M.B. Smith, and C.C. Pantelides A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs Comput. Chem. Eng. 23 1999 457 478 (Pubitemid 29210308)