Improved relaxations for the parametric solutions of ODEs using differential inequalities

Journal of Global Optimization

Volumn 57, Issue 1, 2013, Pages 143-176

  Scott, Joseph K ^a   Barton, Paul I ^a  

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

Convex relaxations; Global optimization; Optimal control

Indexed keywords

AUXILIARY SYSTEMS; CONCAVE RELAXATIONS; CONVEX RELAXATION; DETERMINISTIC GLOBAL OPTIMIZATION; DIFFERENTIAL INEQUALITIES; ENGINEERING APPLICATIONS; OPTIMAL CONTROLS; PARAMETRIC SOLUTIONS;

ALGORITHMS; GLOBAL OPTIMIZATION; RELAXATION PROCESSES;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 84884279590     PISSN: 09255001     EISSN: 15732916     Source Type: Journal    
DOI: 10.1007/s10898-012-9909-0     Document Type: Article

References (38)

1. Adjiman C.S., Dallwig S., Floudas C.A., Neumaier A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs - I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137-1158 (1998)
2. Aubin J.P.: Viability Theory. Birkhauser, Boston (1991)
3. Banga J., Seider W.: Global optimization of chemical processes using stochastic algorithms. In: Floudas, C., Pardalos, P. (eds.) State of the Art in Global Optimization: Computational Methods and Applications, Kluwer, Dordrecht (1996)
4. Bompadre A., Mitsos A.: Convergence rate of McCormick relaxations. J. Glob. Optim. 52(1), 1-28 (2012)
5. Carrasco E., Banga J.: Dynamic optimization of batch reactors using adaptive stochastic algorithms. Ind. Eng. Chem. Res. 36(6), 2252-2261 (1997) (Pubitemid 127697624)
6. Castiglione F., Piccoli B.: Cancer immunotherapy, mathematical modeling and optimal control. J. Theor. Biol. 247, 723-732 (2007) (Pubitemid 47081044)
7. Cizniar M., Podmajersky M., Hirmajer T., Fikar M., Latifi A.M.: Global optimization for parameter estimation of differential-algebraic systems. Chem. Pap. 63(3), 274-283 (2009)
8. Cohen S.D., Hindmarsh A.C.: CVODE, a stiff/nonstiff ODE solver in C. Comput. Phys. 10(2), 138-143 (1996)
9. Esposito W.R., Floudas C.A.: Global optimization for the parameter estimation of differential-algabraic systems. Ind. Eng. Chem. Res. 39, 1291-1310 (2000) (Pubitemid 30265962)
10. Filippov A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht (1988)
11. Harrison, G.W.: Dynamic models with uncertain parameters. In: Avula, X. (eds.) Proceedings of the 1st International Conference on Mathematical Modeling, vol. 1, pp. 295-304 (1977)
12. Huang H., Adjiman C.S., Shah N.: Quantitative framework for reliable safety analysis. AIChE J. 48(1), 78-96 (2002) (Pubitemid 36858928)
13. Khalil K.H.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)
14. Lin Y., Stadtherr M.A.: Deterministic global optimization for parameter estimation of dynamic systems. Ind. Eng. Chem. Res. 45, 8438-8448 (2006) (Pubitemid 46057394)
15. Lin Y., Stadtherr M.A.: Deterministic global optimization of nonlinear dynamic systems. AIChE J. 53(4), 866-875 (2007) (Pubitemid 46536319)
16. Luus R., Dittrich J., Keil F.: Multiplicity of solutions in the optimization of a bifunctional catalyst blend in a tubular reactor. Can. J. Chem. Eng. 70, 780-785 (1992) (Pubitemid 23622429)
17. Martin R.: Optimal control drug scheduling of cancer chemotherapy. Automatica 28(6), 1113-1123 (1992) (Pubitemid 23587963)
18. McCormick G.P.: Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems. Math. Program. 10, 147-175 (1976)
19. Mitsos A., Chachuat B., Barton P.I.: McCormick-based relaxations of algorithms. SIAM J. Optim. 20(2), 573-601 (2009)
20. Moore R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)
21. Neher M., Jackson K.R., Nedialkov N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45(1), 236-262 (2007) (Pubitemid 351321966)
22. Papamichail I., Adjiman C.S.: A rigorous global optimization algorithm for problems with ordinary differential equations. J. Glob. Optim. 24(1), 1-33 (2002) (Pubitemid 34868033)
23. Papamichail I., Adjiman C.S.: Global optimization of dynamic systems. Comput. Chem. Eng. 28, 408-415 (2004) (Pubitemid 38156264)
24. Park T., Barton P.: State event location in differential-algebraic models. ACM Trans. Model. Comput. Simul. 6(2), 137-165 (1996)
25. Sahlodin A.M., Chachuat B.: Convex/concave relaxations of parametric ODEs using Taylor models. Comp. Chem. Eng. 35, 844-857 (2011)
26. Sahlodin A.M., Chachuat B.: Discretize-then-relax approach for convex/concave relaxations of the solutions of parametric ODEs. Appl. Numer. Math. 61, 803-820 (2011)
27. Scott J.K., Barton P.I.: Tight, efficient bounds on the solutions of chemical kinetics models. Comput. Chem. Eng. 34, 717-731 (2010)
28. Scott, J.K., Barton, P.I.: Bounds on the reachable sets of nonlinear control systems (2011, submitted)
29. Scott, J.K., Chachuat, B., Barton, P.I.: Nonlinear convex and concave relaxations for the solutions of parametric ODEs. Optim. Control Appl. Methods (2012, in press). doi: 10.1002/oca.2014
30. Scott J.K., Stuber M.D., Barton P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51, 569-606 (2011). doi: 10.1007/s10898-011-9664-7
31. Singer A.B., Barton P.I.: Global solution of optimization problems with parameter-embedded linear dynamic systems. J. Optim. Theory Appl. 121, 613-646 (2004) (Pubitemid 39168027)
32. Singer A.B., Barton P.I.: Bounding the solutions of parameter dependent nonlinear ordinary differential equations. SIAM J. Sci. Comput. 27, 2167-2182 (2006) (Pubitemid 44784233)
33. Singer A.B., Barton P.I.: Global dynamic optimization for parameter estimation in chemical kinetics. J. Phys. Chem. A 110(3), 971-976 (2006)
34. Singer A.B., Barton P.I.: Global optimization with nonlinear ordinary differential equations. J. Glob. Optim. 34, 159-190 (2006) (Pubitemid 43269381)
35. Srinivasan B., Palanki S., Bonvin D.: Dynamic optimization of batch processes - I. characterization of the nominal solution. Comp. Chem. Eng. 27(1), 1-26 (2003) (Pubitemid 36077352)
36. Szarski J.: Differential Inequalities. Polish Scientific Publishers, Warszawa (1965)
37. Taylor J.W., Ehlker G., Carstensen H.H., Ruslen L., Field R.W., Green W.H.: Direct measurement of the fast, reversible addition of oxygen to cyclohexadienyl radicals in nonpolar solvents. J. Phys. Chem. A 108, 7193-7203 (2004)
38. Walter W.: Differential and Integral Inequalities. Springer, New York (1970)