Computational Optimization and Applications

, Volume 70, Issue 3, pp 641–675 | Cite as

Combinatorial optimal control of semilinear elliptic PDEs

  • Christoph Buchheim
  • Renke Kuhlmann
  • Christian Meyer


Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved.


Optimal control Discrete optimization Outer approximation Submodularity 


  1. 1.
    Alibert, J.-J., Raymond, J.-P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim. 18, 235–250 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avraam, M., Shah, N., Pantelides, C.: Modelling and optimisation of general hybrid systems in the continuous time domain. Comput. Chem. Eng. 22(Supplement 1), S221–S228 (1998)CrossRefGoogle Scholar
  3. 3.
    Balakrishna, S., Biegler, L.T.: A unified approach for the simultaneous synthesis of reaction, energy, and separation systems. Ind. Eng. Chem. Res. 32, 1372–1382 (1993)CrossRefGoogle Scholar
  4. 4.
    Bansal, V., Sakizlis, V., Ross, R., Perkins, J.D., Pistikopoulos, E.N.: New algorithms for mixed-integer dynamic optimization. Comput. Chem. Eng. 27, 647–668 (2003)CrossRefGoogle Scholar
  5. 5.
    Baumann, F., Berckey, S., Buchheim, C.: Exact algorithms for combinatorial optimization problems with submodular objective functions. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization-Festschrift for Martin Grötschel, pp. 271–294. Springer, Berlin (2013)CrossRefGoogle Scholar
  6. 6.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boehme, T.J., Schori, M., Frank, B., Schultalbers, M., Lampe B.: Solution of a hybrid optimal control problem for parallel hybrid vehicles subject to thermal constraints. In: 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), IEEE, pp. 2220–2226 (2013)Google Scholar
  8. 8.
    Bonami, P., Biegler, L., Conn, A., Cornuéjols, G., Grossmann, I., Laird, C., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–204 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24, 1309–1318 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chandra, R.: Partial differential equations constrained combinatorial optimization on an adiabatic quantum computer. Master’s thesis, Purdue University (2013)Google Scholar
  12. 12.
    De Santis, M., Di Pillo, G., Lucidi, S.: An active set feasible method for large-scale minimization problems with bound constraints. Comput. Optim. Appl. 53, 395–423 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45, 1937–1953 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dimitriadis, V.D., Pistikopoulos, E.N.: Flexibility analysis of dynamic systems. Ind. Eng. Chem. Res. 34, 4451–4462 (1995)CrossRefGoogle Scholar
  15. 15.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization—Eureka, You Shrink!, vol. 2570 of LNCS, pp. 11–26. Springer (2003)Google Scholar
  17. 17.
    Fügenschuh, A., Geissler, B., Martin, A., Morsi, A.: The transport PDE and mixed-integer linear programming. In: Barnhart, C., Clausen, U., Lauther, U., Möhring, R.H. (eds.) Models and Algorithms for Optimization in Logistics, no. 09261 in Dagstuhl Seminar Proceedings, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany (2009)Google Scholar
  18. 18.
    Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics. Elsevier, Amsterdam (1991)zbMATHGoogle Scholar
  19. 19.
    Geissler, B., Kolb, O., Lang, J., Leugering, G., Martin, A., Morsi, A.: Mixed integer linear models for the optimization of dynamical transport networks. Math. Methods Oper. Res. 73, 339–362 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gerdts, M.: A variable time transformation method for mixed-integer optimal control problems. Optim. Control Appl. Methods 27, 169–182 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics. SIAM, Philadelphia (1985)zbMATHGoogle Scholar
  22. 22.
    Haller-Dintelmann, R., Meyer, C., Rehberg, J., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60, 397–428 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hante, F.M., Sager, S.: Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55, 197–225 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hintermüller, M., Kunisch, K.: PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim. 20, 1133–1156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Incropera, F., Witt, D.D.: Fundamentals of Heat and Mass Transfer. Wiley, Chichester (1985)Google Scholar
  26. 26.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, vol. 31. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Kirches, C., Lenders, F.: Approximation properties and tight bounds for constrained mixed-integer optimal control. Technical report, Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University (2016)Google Scholar
  28. 28.
    Kirches, C., Sager, S., Bock, H.G., Schlöder, J.P.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Methods 31, 137–153 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming: The State of the Art (Bonn, 1982), pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  30. 30.
    Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37, 51–85 (2008)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22, 871–899 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mohideen, M.J., Perkins, J.D., Pistikopoulos, E.N.: Optimal design of dynamic systems under uncertainty. AIChE J. 42, 2251–2272 (1996)CrossRefGoogle Scholar
  33. 33.
    Quesada, I., Grossmann, I.: An LP/NLP based branched and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)CrossRefGoogle Scholar
  34. 34.
    Sager, S., Bock, H., Diehl, M.: The integer approximation error in mixed-integer optimal control. Math. Program. 133, 1–23 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sager, S., Jung, M., Kirches, C.: Combinatorial integral approximation. Math. Methods Oper. Res. 73, 363–380 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schiela, A., Wollner, W.: Barrier methods for optimal control problems with convex nonlinear gradient state constraints. SIAM J. Optim. 21, 269–286 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Till, J., Engell, S., Panek, S., Stursberg, O.: Applied hybrid system optimization: an empirical investigation of complexity. Control Eng. Practice 12, 1291–1303 (2004)CrossRefGoogle Scholar
  38. 38.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate studies in Mathematics. American Mathematical Society, Providence (2010)CrossRefzbMATHGoogle Scholar
  39. 39.
    von Stryk, O., Glocker, M.: Decomposition of mixed-integer optimal control problems using branch and bound and sparse direct collocation. In: Engell, S., Kowalewski, S., Zaytoon, J. (eds.) Proceedings of ADPM 2000—The 4th International Conference on Automation of Mixed Processes: Hybrid Dynamic Systems, Dortmund, pp. 99–104, Sept. 2000Google Scholar
  40. 40.
    Zhang, P., Romero, D., Beck, J., Amon, C.: Solving wind farm layout optimization with mixed integer programming and constraint programming. In: Gomes, C., Sellmann, M. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, vol. 7874 of Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, pp. 284–299 (2013)Google Scholar

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Authors and Affiliations

  1. 1.Fakultät für MathematikTU DortmundDortmundGermany

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