Abstract
Large scale optimization strategies have evolved considerably over the past two decades. Currently, continuous variable optimization problems (nonlinear programs) are solved on-line for steady state refinery models with several hundred thousand variables. Moreover, efficient NLP strategies have been developed for dynamic optimization problems. Still, to take the next step, on-line optimization of large dynamic chemical processes, a number of limitations and research challenges must be overcome.
Many of the advances in NLP algorithms have taken place by recognizing and exploiting the framework of Successive Quadratic Programming (SQP) algorithms. These are extensions of Newton type methods for converging to the solution of the KKT (optimality) conditions of the optimization problem. Because of this, fast convergence can be expected and a number of standard devices can be added to stabilize the algorithm to converge from poor starting points. Limitations of these Newton-based methods are also well-known. They experience difficulties in the presence of ill conditioning and extreme nonlinearities. Also, for optimization algorithms, nonconvexity can also lead to a number of difficulties and there is a need for software that allows exploitable structures for specific problem classes.
A number of innovations in algorithm design and problem formulation address these issues and greatly improve performance. As a result, very fast NLP algorithms can be derived for data reconciliation, parameter estimation, nonlinear model predictive control and dynamic optimization. Moreover, inequality constraints and variable bounds can be treated through advances in interior point strategies. These methods preserve the particular problem structure and scale well in performance for large-scale problems with many constraints.
Finally, the ability to solve nonlinear programs quickly also allows us to consider more challenging problems. These include the extension to solving nonconvex NLPs globally and the ability to assess the solution’s tolerance to uncertainty by considering features of nonlinear programming sensitivity analysis and robust optimization through flexibility analysis and design.
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References
Albuquerque, A. and L. T. Biegler, “Decomposition algorithms for on-line estimation with nonlinear models,” Comp. Chem. Engr 19, p. 1031 (1995)
Albuquerque, A. and L. T. Biegler, Decomposition algorithms for on-line estimation with nonlinear DAE models, Comp. Chem. Engr 21, p. 283 (1997)
[3] Albuquerque, J V. Gopal, G. Staus, L. T. Biegler, and B. E. Ydstie, “Interior Point SQP Strategies For Structured Process Optimization Problems” Process Syste ms Engineering ‘87
Ascher, U. M R.M.M. Mattheij and R.D. Russel, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall Englewood Cliffs,NJ. (1988)
Badgwell, T “Robust model predictive control for nonlinear plants,” presented at Annual AIChE meeting, Chicago, IL (1996)
Beard, R. W G. N. Saridis, and J. T. Wen, “Galerkin Approximations of the Generalized Hamilton-Jacobi-Bellman Equation,” Automatica, 33, 12, P. 2159 (1997)
Biegler, L. T J. Nocedal and C. Schmid, A Reduced Hessian Method for Large-Scale Constrained Optimization, SIAM J. Optimization, 5, p. 314 (1995)
Biegler, L. T Advances in Nonlinear Programming Concepts for Process Control,“ J. Proc. Control, to appear (1998)
Bock, H. G “Recent advances in parameter identification techniques,” in P. Deuflhard and E. Hairer (eds.), Numerical Treatment of Inverse Problems, Birkhäuser, Heidelberg (1983)
deBoor, C. W and R. Weiss, “SOLVEBLOK: A package for solving ABD linear ssytems,” ACM TOMS, 6,p. 80 (1980)
S. Boyd, C. Crusius, and A. Hansson, Control Applications of Nonlinear Convex Programming. J. Process Control, to appear, 1997
Byrd, R J-C. Gilbert and J. Nocedal, “A trust region method based on interior point methods for nonlinear programming,” Technical Report OTC 96/02, Northwestern University (1996)
Byrd, R M. E. Hribar and J. Nocedal, “An interior point algorithm for large-scale nonlinear programming,” Technical Report OTC 97/05, Northwestern University (1997)
Cervantes, A. and L. T. Biegler, “Large-Scale DAE Optimization Using a Simultaneous NLP Formulation,” AIChE J., to appear (1998)
Dennis, J. and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia (1996)42 L. T. Biegler
Downs, J.J. and E.F. Vogel, A Plant-Wide Industrial Process Control Problem, Computers Chem. Eng Vol 17(3), p. 245–255 (1993)
Dunn, J. C. and D. Bertsekas, “Efficient dynamic programming implementations of Newton’s method for unconstrained optimal control problems,” J. Opt. Theo. Appl., 62, p. 23 (1989)
Epperly, T. G. W and E. N. Pistikopoulos, “A reduced space branch and bound algorithm for global optimization,” J. Global Optimization, 11, p. 287 (1997)
Fletcher, R Practical Methods for Optimization, Wiley, Chichester (1987)
Floudas, C. A “Deterministic Global Optimization in Design, Control, and Computational Chemistry,” in Large Scale Optimization with Applications, Volume 93: Part II: Optimal Design and Control, IMA Volumes in Mathematics and Applications, Springer Verlag, New York (1997)
Copal, V PhD thesis, Carnegie Mellon University, Pittsburgh, PA, 1997.
Grossmann, I. E. (ed.) Global Optimization in Engineering Design,Kluwer, Netherlands (1996)
S-P. Han. “A globally convergent method for nonlinear programming.” JOTA, 22(3):297–310, 1977.
deHoog, R and R. M. Mattheij, “On the conditioning of multipoint and integral boundary value problems,” SIAM J. Math. Anal., 20, 1, p. 200 (1989)
Horst, R. and H. Tuy, Global Optimization Springer Verlag, Berlin (1996)
Kassmann, D. E and T. A. Badgwell, “Interior Point Methods in Robust Model Predictive Control,” presented at Annual AIChE Meeting, Los Angeles, CA (1997)
Keerthi, S.S E.G. Gilbert, “Optimal Infinite-Horizon Feedback Laws for General Class of Constrained Discrete-Time Systems: Stability and Moving-Horizon Approximations”, IEEE Trans. Auto. Cont., 57(2), 265–293 (1988)
Kleinman, D. L “An Easy Way to Stabilize a Linear Constant System,” IEEE Trans. Aut. Cont AC-15, p. 692 (1970)
Kothare, M V. Balakrishnan and M. Morari, “Robust Constrained Model Predictive Control using Linear Matrix Inequalities,” Automatica, 32, 10, p. 1361 (1996)
Kwon, W. H. and A. E. Pearson, “On Feadback Stabilization of Time Varying Discrete Linear Systems,” IEEE Trans. Aut. Cont AC-23, p. 838 (1977)
Lalee, M J. Nocedal and T. Platenga, “On the implementation of an algorithm for large-scale equality constrained optimization,” SIAM J. Opt., to appear (1998)
Li, W-C. and L. T. Biegler, “Multistep, Newton-type Control Strategies for Constrained Nonlinear Processes,” Chemical Engineering Research and Design,67, p. 562 (1989)
Ljung, L (1987), System Identification, Theory for the User, Prentice Hall, Inc Englewood Cliffs, New Jersey.
Mayne, D. Q, “Nonlinear model predictive conrol: an assessment,” Proceedings of CPC-V, (1996)
Michalska, H D.Q. Mayne (1993). “Robust Receding Horizon Control of Constrained Nonlinear Systems”, IEEE Trans. Auto. Cont., 38(11), 1623–1633.
McCormick, G.P “Computability of Global Solutions to Factorable Nonconvex Programs: Part I: Convex underestimator problems”, Math. Prog.,Vol. 10, p.147 (1976)
Mehrotra, S Quadratic convergence in a primal dual method, Math. Oper. Res 15, p. 741 (1993)
Mohideen, M. J PhD Thesis, Imperial College, London (1996)
Oliveira, N.M.C L.T. Siegler “Newton-type Algorithms for Nonlinear Process Control. Algorithm and Stability Results”, Automatica,31, 2, p. 281 (1995)
Pistikopoulos, E “Uncertainty in Process Design and Operations,” Comp. Chem. Engr., 19, p. S553 (1995)
Rao, C J. B. Rawlings and S. Wright, “Application of Interior Point Methods to Model Predictive Control,” J. Opt. Theo. Applies., to appear (1998)
Rawlings, J.B K.R. Muske (1993). “The Stability of Constrained Receding Horizon Control”, IEEE Trans. Auto. Cont., 38(10), 1512–1516.
Santos, L.O N. Oliveira and L.T. Siegler, “Reliable and Efficient Optimization Strategies for Nonlinear Model Predictive Control,” Proc. Fourth IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes (DYCORD ‘85), p. 33 (1995).
L. O. Santos and L. T. Siegler, “A Tool to Analyze Robust Stability for Model Predictive Controllers,” J. Process Control, to appear (1998)
Scockaert, P. and J. B. Rawlings, “Optimization formulations for model predictive control,” Proceedings of IMA Workshop on Large Scale Optimization (1997)
Scockaert, P D. Q. Mayne and J. B. Rawlings, “Suboptimal model predictive control,” submitted for publication (1997)
Smith, E. M. B and C. Pantelides, “Global Optimisation of Nonconvex MINLPs,” Comp. Chem. Engr., 21, p. S791—S796 (1997)
Staus, G. H L. T. Siegler and B. E. Ydstie, “Global Optimization for Identification,” submitted for publication (1998)
Steihaug, T “The conjugate gradient method and trust regions in large scale optimization,” SIAM J. Num. Anal., 20, p. 626 (1983)
Tanartkit, P L.T. Siegler, “Stable Decomposition for Dynamic Optimization”, I& EC Research, 34, p. 1253 (1995)
Ternet, D PhD thesis, Carnegie Mellon University, Pittsburgh, PA, 1997.
Wright, S “Applying New Optimization Algorithms to Model Predictive Control,” Proceedings of CPC-V, (1996)
Wright, S Primal-Dual Interior Point Methods, SIAM, Philadelphia (1997)
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Biegler, L.T. (2000). Efficient Solution of Dynamic Optimization and NMPC Problems. In: Allgöwer, F., Zheng, A. (eds) Nonlinear Model Predictive Control. Progress in Systems and Control Theory, vol 26. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8407-5_13
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DOI: https://doi.org/10.1007/978-3-0348-8407-5_13
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