Abstract
Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, and vanishing constraints. In addition to the tightness of the respective relaxations, we also address the issue of constraint qualification and the behavior of computational methods for some formulations. As a benchmark, we formulate a truck cruise control problem with logical implications resulting from gear-choice specific constraints. We provide this benchmark problem in AMPL format along with different realistic scenarios. Computational results for this benchmark are used to investigate feasibility gaps, integer feasibility gaps, quality of local solutions, and well-behavedness of the presented reformulations of the benchmark problem. Vanishing constraints give the most satisfactory results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abichandani, P., Benson, H., Kam, M.: Multi-vehicle path coordination under communication constraints. In: American Control Conference, pp. 650–656 (2008)
Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math. Program., Ser. A 114, 69–99 (2008)
Anitescu, M., Tseng, P., Wright, S.: Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties. Math. Program., Ser. A 110, 337–371 (2007)
Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)
Bär, V.: Ein Kollokationsverfahren zur numerischen Lösung allgemeiner Mehrpunktrandwert aufgaben mit Schalt- und Sprungbedingungen mit Anwendungen in der optimalen Steuerung und der Parameteridentifizierung. Diploma thesis, Rheinische Friedrich-Wilhelms-Universität zu Bonn (1983)
Barton, P.: The modelling and simulation of combined discrete/continuous processes. Ph.D. thesis, Department of Chemical Engineering, Imperial College of Science, Technology and Medicine, London (1992)
Baumrucker, B., Biegler, L.: MPEC strategies for optimization of a class of hybrid dynamic systems. J. Process Control 19(8), 1248–1256 (2009)
Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. In: Iserles, A. (ed.) Acta Numerica 2013, vol. 22. Cambridge University Press, Cambridge (2013). www.optimization-online.org/DB_HTML/2012/12/3698.html
Betts, J.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)
Biegler, L.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)
Biegler, L.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. Series on Optimization. SIAM, Philadelphia (2010)
Bock, H., Longman, R.: Computation of optimal controls on disjoint control sets for minimum energy subway operation. Adv. Astronaut. Sci. 50, 949–972 (1985)
Bock, H., Plitt, K.: A Multiple Shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242–247. Pergamon Press, Budapest (1984)
Burgschweiger, J., Gnädig, B., Steinbach, M.: Nonlinear programming techniques for operative planning in large drinking water networks. Open Appl. Math. J. 3, 1–16 (2009)
Ceria, S., Soares, J.: Convex programming for disjunctive optimization. Math. Program. 86, 595–614 (1999)
Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)
Fang, H., Leyffer, S., Munson, T.: A pivoting algorithm for linear programs with complementarity constraints. Optim. Methods Softw. 87, 89–114 (2012)
Ferris, M., Kanzow, C.: Complementarity and related problems: a survey (1998)
Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)
Fletcher, R., de la Maza, E.S.: Nonlinear programming and nonsmooth optimization by successive linear programming. Math. Program. 43(3), 235–256 (1989)
Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optim. Methods Softw. 19(1), 15–40 (2004)
Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program., Ser. A 106, 225–236 (2006)
Fügenschuh, A., Herty, M., Klar, A., Martin, A.: Combinatorial and continuous models for the optimization of traffic flows on networks. SIAM J. Optim. 16(4), 1155–1176 (2006)
Fukushima, M., Qi, L. (eds.): Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Kluwer Academic, Dordrecht (1999)
Gerdts, M.: Solving mixed-integer optimal control problems by Branch&Bound: a case study from automobile test-driving with gear shift. Optim. Control Appl. Methods 26, 1–18 (2005)
Gerdts, M.: A variable time transformation method for mixed-integer optimal control problems. Optim. Control Appl. Methods 27(3), 169–182 (2006)
Gerdts, M., Sager, S.: Mixed-integer DAE optimal control problems: necessary conditions and bounds. In: Biegler, L., Campbell, S., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints, pp. 189–212. SIAM, Philadelphia (2012)
Gill, P., Murray, W., Saunders, M.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12, 979–1006 (2002)
Göttlich, S., Herty, M., Kirchner, C., Klar, A.: Optimal control for continuous supply network models. Netw. Heterog. Media 1(4), 675–688 (2007)
Gräber, M., Kirches, C., Bock, H., Schlöder, J., Tegethoff, W., Köhler, J.: Determining the optimum cyclic operation of adsorption chillers by a direct method for periodic optimal control. Int. J. Refrig. 34(4), 902–913 (2011)
Grossmann, I.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)
Gugat, M., Herty, M., Klar, A., Leugering, G.: Optimal control for traffic flow networks. J. Optim. Theory Appl. 126(3), 589–616 (2005)
Günlük, O., Linderoth, J.: Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Program. 124(1–2), 183–205 (2010)
Hante, F., Sager, S.: Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55(1), 197–225 (2013)
Hellström, E., Ivarsson, M., Aslund, J., Nielsen, L.: Look-ahead control for heavy trucks to minimize trip time and fuel consumption. Control Eng. Pract. 17, 245–254 (2009)
Hoheisel, T.: Mathematical programs with vanishing constraints. Ph.D. thesis, Julius-Maximilians-Universität Würzburg (2009)
Jung, M.N., Reinelt, G., Sager, S.: The Lagrangian relaxation for the combinatorial integral approximation problem. Optim. Methods Softw. (2012, submitted). www.optimization-online.org/DB_HTML/2012/02/3354.html
Kawajiri, Y., Biegler, L.: A nonlinear programming superstructure for optimal dynamic operations of simulated moving bed processes. Ind. Eng. Chem. Res. 45(25), 8503–8513 (2006)
Kirches, C.: Fast Numerical Methods for Mixed-Integer Nonlinear Model-Predictive Control. Advances in Numerical Mathematics. Springer Vieweg, Wiesbaden (2011)
Kirches, C., Sager, S., Bock, H., Schlöder, J.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Methods 31(2), 137–153 (2010)
Kirches, C., Bock, H., Schlöder, J., Sager, S.: Block structured quadratic programming for the direct multiple shooting method for optimal control. Optim. Methods Softw. 26(2), 239–257 (2011)
Kirches, C., Bock, H., Schlöder, J., Sager, S.: A factorization with update procedures for a KKT matrix arising in direct optimal control. Math. Program. Comput. 3(4), 319–348 (2011)
Kirches, C., Wirsching, L., Bock, H., Schlöder, J.: Efficient direct multiple shooting for nonlinear model predictive control on long horizons. J. Process Control 22(3), 540–550 (2012)
Leineweber, D., Bauer, I., Schäfer, A., Bock, H., Schlöder, J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II). Comput. Chem. Eng. 27, 157–174 (2003)
Leyffer, S.: Complementarity constraints as nonlinear equations: theory and numerical experience. In: Optimization with Multivalued Mappings: Theory, Applications, and Algorithms, pp. 169–208. Springer, Berlin (2006)
Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optim. 17(1), 52–77 (2006)
Leyffer, S., Munson, T.: A globally convergent filter method for MPECs. Preprint ANL/MCS-P1457-0907, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USA (2007)
Logist, F., Sager, S., Kirches, C., van Impe, J.: Efficient multiple objective optimal control of dynamic systems with integer controls. J. Process Control 20(7), 810–822 (2010)
Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Math. Program. 105, 563–582 (2006)
Oldenburg, J., Marquardt, W.: Disjunctive modeling for optimal control of hybrid systems. Comput. Chem. Eng. 32(10), 2346–2364 (2008)
Prata, A., Oldenburg, J., Kroll, A., Marquardt, W.: Integrated scheduling and dynamic optimization of grade transitions for a continuous polymerization reactor. Comput. Chem. Eng. 32, 463–476 (2008)
Raghunathan, A., Biegler, L.: Mathematical programs with equilibrium constraints (MPECs) in process engineering. Comput. Chem. Eng. 27, 1381–1392 (2003)
Raghunathan, A., Diaz, M., Biegler, L.: An MPEC formulation for dynamic optimization of distillation operations. Comput. Chem. Eng. 28, 2037–2052 (2004)
Ralph, D., Wright, S.J.: Some properties of regularization and penalization schemes for MPECs. Optim. Methods Softw. 19, 527–556 (2004)
Sager, S.: MIOCP benchmark site. mintoc.de
Sager, S.: Numerical Methods for Mixed-Integer Optimal Control Problems. Der Andere Verlag, Tönning (2005)
Sager, S.: Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J. Process Control 19(8), 1238–1247 (2009)
Sager, S.: On the integration of optimization approaches for mixed-integer nonlinear optimal control. Habilitation, University of Heidelberg (2011)
Sager, S.: A benchmark library of mixed-integer optimal control problems. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 631–670. Springer, Berlin (2012)
Sager, S., Reinelt, G., Bock, H.: Direct methods with maximal lower bound for mixed-integer optimal control problems. Math. Program. 118(1), 109–149 (2009)
Sager, S., Bock, H., Diehl, M.: The integer approximation error in mixed-integer optimal control. Math. Program., Ser. A 133(1–2), 1–23 (2012)
Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001)
Scholtes, S.: Nonconvex structures in nonlinear programming. Oper. Res. 52(3), 368–383 (2004)
Sherali, H.: RLT: a unified approach for discrete and continuous nonconvex optimization. Ann. Oper. Res. 149, 185–193 (2007)
Sonntag, C., Stursberg, O., Engell, S.: Dynamic optimization of an industrial evaporator using graph search with embedded nonlinear programming. In: Proceedings of the 2nd IFAC Conference on Analysis and Design of Hybrid Systems (ADHS), pp. 211–216 (2006)
Stein, O., Oldenburg, J., Marquardt, W.: Continuous reformulations of discrete-continuous optimization problems. Comput. Chem. Eng. 28(10), 3672–3684 (2004)
Stubbs, R., Mehrotra, S.: Generating convex polynomial inequalities for mixed 0–1 programs. J. Glob. Optim. 24, 311–332 (2002)
Terwen, S., Back, M., Krebs, V.: Predictive powertrain control for heavy duty trucks. In: Proceedings of IFAC Symposium in Advances in Automotive Control, Salerno, Italy, pp. 451–457 (2004)
Wächter, A., Biegler, L.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Jung, M.N., Kirches, C., Sager, S. (2013). On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-38189-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38188-1
Online ISBN: 978-3-642-38189-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)