Combinatorial integral approximation

  • Sebastian Sager
  • Michael Jung
  • Christian Kirches
Original Article


We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process. The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms cplex on a numerical case study and hence further improves efficiency of our novel method.


MINLP MIOCP MILP Optimal control Integer programming 

Mathematics Subject Classification (2000)

90C11 90C30 49J15 90C57 


  1. Abhishek K, Leyffer S, Linderoth J (2006) Filmint: an outer-approximation-based solver for nonlinear mixed integer programs. Preprint ANL/MCS-P1374-0906, Argonne National Laboratory, Mathematics and Computer Science DivisionGoogle Scholar
  2. Applegate D, Bixby R, Chvátal V, Cook W, Espinoza D, Goycoolea M, Helsgaun K (2009) Certification of an optimal TSP tour through 85, 900 cities. Oper Res Lett 37(1): 11–15MathSciNetzbMATHCrossRefGoogle Scholar
  3. Belotti P, Lee J, Liberti L, Margot F, Waechter A (2009) Branching and bounds tightening techniques for non-convex MINLP. Optim Methods Softw 24: 597–634MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bixby RE, Fenelon M, Gu Z, Rothberg E, Wunderling R (2004) Mixed-integer programming: a progress report. In: The sharpest cut: the impact of manfred padberg and his work. SIAMGoogle Scholar
  5. Bonami P, Biegler L, Conn A, Cornuéjols G, Grossmann I, Laird C, Lee J, Lodi A, Margot F, Sawaya N, Wächter A (2009) An algorithmic framework for convex mixed integer nonlinear programs. Discr Optim 5(2): 186–204CrossRefGoogle Scholar
  6. Bonami P, Cornuejols G, Lodi A, Margot F (2009) Feasibility pump for mixed integer nonlinear programs. Math Program 199: 331–352MathSciNetCrossRefGoogle Scholar
  7. Bonami P, Kilinc M, Linderoth J (2009) Algorithms and software for convex mixed integer nonlinear programs. Technical Report 1664, University of WisconsinGoogle Scholar
  8. Burgschweiger J, Gnädig B, Steinbach M (2008) Optimization models for operative planning in drinking water networks. Optim Eng 10(1): 43–73 Online firstCrossRefGoogle Scholar
  9. Christof T, Löbel A PORTA—polyhedron representation transformation algorithm PORTA Homepage
  10. Christof T, Reinelt G (1996) Combinatorial optimization and small polytopes. TOP 4(1): 1–53MathSciNetzbMATHCrossRefGoogle Scholar
  11. Floudas C, Akrotirianakis I, Caratzoulas S, Meyer C, Kallrath J (2005) Global optimization in the 21st century: advances and challenges. Comput Chem Eng 29(6): 1185–1202CrossRefGoogle Scholar
  12. Grossmann I (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3: 227–252MathSciNetzbMATHCrossRefGoogle Scholar
  13. Kameswaran S, Biegler L (2006) Simultaneous dynamic optimization strategies: recent advances and challenges. Comput Chem Eng 30: 1560–1575CrossRefGoogle Scholar
  14. Lee J, Leung J, Margot F (2004) Min-up/ min-down polytopes. Discr Optim 1: 77–85MathSciNetzbMATHCrossRefGoogle Scholar
  15. Leineweber D, Bauer I, Bock H, Schlöder J (2003) An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization, Part I: theoretical aspects. Comput Chem Eng 27: 157–166CrossRefGoogle Scholar
  16. Margaliot M (2007) A counterexample to a conjecture of Gurvits on switched systems. IEEE Trans Automat Control 52(6): 1123–1126MathSciNetCrossRefGoogle Scholar
  17. Sager S (2005) Numerical methods for mixed–integer optimal control problems. Der andere Verlag, Tönning, Lübeck, Marburg ISBN 3-89959-416-9. Available at
  18. Sager S (2009) Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J Process Control 19(8): 1238–1247CrossRefGoogle Scholar
  19. Sager S, Reinelt G, Bock H (2009) Direct methods with maximal lower bound for mixed-integer optimal control problems. Math Program 118(1): 109–149MathSciNetzbMATHCrossRefGoogle Scholar
  20. Sager S, Bock H, Diehl M (2011) The integer approximation error in mixed-integer optimal control. Accepted by Mathematical Programming doi: 10.1007/s10107-010-0405-3
  21. Sharon Y, Margaliot M (2007) Third-order nilpotency, finite switchings and asymptotic stability. J Differ Equ 233: 135–150MathSciNetCrossRefGoogle Scholar
  22. Tawarmalani M, Sahinidis N (2002) Convexification and global optimization in continuous and mixed- integer nonlinear programming: theory, algorithms, software, and applications. Kluwer, DordrechtzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Sebastian Sager
    • 1
  • Michael Jung
    • 1
  • Christian Kirches
    • 1
  1. 1.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany

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