Abstract
A classical hybrid MIP-CSP approach for solving problems having a logical part and a mixed integer programming part is presented. A Branch and Bound procedure combines an MIP and a SAT solver to determine the optimal solution of a general class of optimization problems. The procedure explores the search tree, by solving at each node a linear relaxation and a satisfiability problem, until all integer variables of the linear relaxation are set to an integer value in the optimal solution. When all integer variables are fixed the procedure switches to the SAT solver which tries to extend the solution taking into account logical constraints. If this is impossible, a “no-good” cut is generated and added to the linear relaxation. We show that the class of problems we consider turns out to be very useful for solving complex optimal control problems for linear hybrid dynamical systems formulated in discrete-time. We describe how to model the “hybrid” dynamics so that the optimal control problem can be solved by the hybrid MIP+SAT solver, and show that the achieved performance is superior to the one achieved by commercial MIP solvers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Marriot, K., Stuckey, P.J.: Programming with constraints: an introduction. MIT Press, Cambridge (1998)
Gu, J., Purdom, P.W., Franco, J., Wah, B.: Algorithms for the satisfiability (SAT) problem: A survey. In: DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 35, pp. 19–151. American Mathematical Society, Providence (1997)
Hooker, J.: Logic-based methods for Optimization. Wiley-Interscience Series (2000)
Bockmayr, A., Kasper, T.: Branch and infer: A unifying framework for integer and finite domain constraint programming. INFORMS Journal on Computing 10(3), 287–300 (1998)
Rodosek, R., Wallace, M., Hajian, M.: A new approach to integrating mixed integer programming and constraint logic programming. Annals of Oper. Res. 86, 63–87 (1997)
Focacci, F., Lodi, A., Milano, M.: Cost-based domain filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (2001)
Harjunkoski, I., Jain, V., Grossmann, I.E.: Hybrid mixed-integer/constraint logic programming strategies for solving scheduling and combinatorial optimization problems. Comp. Chem. Eng. 24, 337–343 (2000)
Large scale integrated supply chain optimisation software. A European Union Funded Project (2003), http://www.liscos.fc.ul.pt/
Xu, X., Antsaklis, P.J.: An approach to switched systems optimal control based on parameterization of the switching instants. In: Proc. IFAC World Congress, Barcelona, Spain (2002)
Lincoln, B., Rantzer, A.: Optimizing linear system switching. In: Proc. 40th IEEE Conf. on Decision and Control, pp. 2063–2068 (2001)
Borrelli, F., Baotic, M., Bemporad, A., Morari, M.: An efficient algorithm for computing the state feedback optimal control law for discrete time hybrid systems. In: Proc. American Contr. Conf., Denver, Colorado (2003)
Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407–427 (1999)
Bemporad, A., Borrelli, F., Morari, M.: Piecewise linear optimal controllers for hybrid systems. In: Proc. American Contr. Conf., pp. 1190–1194, Chicago, IL (June 2000)
Torrisi, F.D., Bemporad, A.: HYSDEL - A tool for generating computational hybrid models. IEEE Transactions on Control Systems Technology 12(2) (March 2004)
Bemporad, A., Giorgetti, N.: A logic-based hybrid solver for optimal control of hybrid systems. In: Proc. 43th IEEE Conf. On Decision and Control, Maui, Hawaii, USA (December 2003)
Williams, H.P.: Model Building in Mathematical Programming, 3rd edn. John Wiley & Sons, Chichester (1993)
Ottosson, G.: Integration of Constraint Programming and Integer Programming for Combinatorial Optimization. PhD thesis, Computing Science Department, Information Technology, Uppsala University, Sweden (2000)
Tsang, E.: Foundations of Constraint Satisfaction. Academic Press, London (1993)
Boyd, S., Vandenberghe, L.: Convex Optimization (2003) (in press), http://www.stanford.edu/char126relaxboyd/cvxbook.html
Hooker, J.N., Osorio, M.A.: Mixed logical/linear programming. Discrete Applied Mathematics 96-97(1-3), 395–442 (1999)
Bemporad, A.: Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form. IEEE Trans. Automatic Control (2003) (in Press)
Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient sat solver. In: 39th Design Automation Conference (June 2001), http://www.ee.princeton.edu/char126relaxchaff/zchaff.php
ILOG, Inc. CPLEX 8.1 User Manual. Gentilly Cedex, France (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bemporad, A., Giorgetti, N. (2004). SAT-Based Branch & Bound and Optimal Control of Hybrid Dynamical Systems. In: Régin, JC., Rueher, M. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2004. Lecture Notes in Computer Science, vol 3011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24664-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-24664-0_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21836-4
Online ISBN: 978-3-540-24664-0
eBook Packages: Springer Book Archive