IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v162y2014i1d10.1007_s10957-013-0426-1.html
   My bibliography  Save this article

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

Author

Listed:
  • Boris Houska

    (Imperial College London)

  • Benoît Chachuat

    (Imperial College London)

Abstract

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram–Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example.

Suggested Citation

  • Boris Houska & Benoît Chachuat, 2014. "Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 208-248, July.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0426-1
    DOI: 10.1007/s10957-013-0426-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-013-0426-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-013-0426-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    4. Eligius M. T. Hendrix & Boglárka G.-Tóth, 2010. "Nonlinear Programming algorithms," Springer Optimization and Its Applications, in: Introduction to Nonlinear and Global Optimization, chapter 5, pages 91-136, Springer.
    5. Joseph Scott & Paul Barton, 2013. "Improved relaxations for the parametric solutions of ODEs using differential inequalities," Journal of Global Optimization, Springer, vol. 57(1), pages 143-176, September.
    6. Agustín Bompadre & Alexander Mitsos & Benoît Chachuat, 2013. "Convergence analysis of Taylor models and McCormick-Taylor models," Journal of Global Optimization, Springer, vol. 57(1), pages 75-114, September.
    7. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    8. Lars Gruene & Willi Semmler, 2002. "Using Dynamic Programming with Adaptive Grid Scheme to Solve Nonlinear Dynamic Models in Economics," Computing in Economics and Finance 2002 99, Society for Computational Economics.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chrysoula D. Kappatou & Dominik Bongartz & Jaromił Najman & Susanne Sass & Alexander Mitsos, 2022. "Global dynamic optimization with Hammerstein–Wiener models embedded," Journal of Global Optimization, Springer, vol. 84(2), pages 321-347, October.
    2. Jason Ye & Joseph K. Scott, 2024. "Modification and improved implementation of the RPD method for computing state relaxations for global dynamic optimization," Journal of Global Optimization, Springer, vol. 89(4), pages 833-861, August.
    3. Ishan Bajaj & M. M. Faruque Hasan, 2020. "Global dynamic optimization using edge-concave underestimator," Journal of Global Optimization, Springer, vol. 77(3), pages 487-512, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Krzysztof S. Targiel & Maciej Nowak & Tadeusz Trzaskalik, 2018. "Scheduling non-critical activities using multicriteria approach," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(3), pages 585-598, September.
    2. Okitonyumbe Y.F., Joseph & Ulungu, Berthold E.-L., 2013. "Nouvelle caractérisation des solutions efficaces des problèmes d’optimisation combinatoire multi-objectif [New characterization of efficient solution in multi-objective combinatorial optimization]," MPRA Paper 66123, University Library of Munich, Germany.
    3. Amit Kumar & Anila Gupta, 2013. "Mehar’s methods for fuzzy assignment problems with restrictions," Fuzzy Information and Engineering, Springer, vol. 5(1), pages 27-44, March.
    4. Monica Motta & Caterina Sartori, 2020. "Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 44-71, April.
    5. Chenchen Wu & Dachuan Xu & Donglei Du & Wenqing Xu, 2016. "An approximation algorithm for the balanced Max-3-Uncut problem using complex semidefinite programming rounding," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1017-1035, November.
    6. Uzma Ashraf & Hassan Ali & Muhammad Nawaz Chaudry & Irfan Ashraf & Adila Batool & Zafeer Saqib, 2016. "Predicting the Potential Distribution of Olea ferruginea in Pakistan incorporating Climate Change by Using Maxent Model," Sustainability, MDPI, vol. 8(8), pages 1-11, July.
    7. World Bank, 2003. "Argentina : Reforming Policies and Institutions for Efficiency and Equity of Public Expenditures," World Bank Publications - Reports 14637, The World Bank Group.
    8. Brown, Jeffrey R., 2001. "Private pensions, mortality risk, and the decision to annuitize," Journal of Public Economics, Elsevier, vol. 82(1), pages 29-62, October.
    9. Mark Christensen, 2007. "What We Might Know (But Aren't Sure) About Public-Sector Accrual Accounting," Australian Accounting Review, CPA Australia, vol. 17(41), pages 51-65, March.
    10. Wong, Patricia J.Y., 2015. "Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 908-930.
    11. Norma M Rantisi & Deborah Leslie, 2021. "In and against the neoliberal state? The precarious siting of work integration social enterprises (WISEs) as counter-movement in Montreal, Quebec," Environment and Planning A, , vol. 53(2), pages 349-370, March.
    12. Brunekreeft, Gert, 2004. "Market-based investment in electricity transmission networks: controllable flow," Utilities Policy, Elsevier, vol. 12(4), pages 269-281, December.
    13. Dias, Luis C. & Lamboray, Claude, 2010. "Extensions of the prudence principle to exploit a valued outranking relation," European Journal of Operational Research, Elsevier, vol. 201(3), pages 828-837, March.
    14. Oloruntoba, Richard, 2010. "An analysis of the Cyclone Larry emergency relief chain: Some key success factors," International Journal of Production Economics, Elsevier, vol. 126(1), pages 85-101, July.
    15. Chein-Shan Liu & Zhuojia Fu & Chung-Lun Kuo, 2017. "Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 9(6), pages 112-123, December.
    16. Schulze, Eberhard & Tillack, Peter & Patlassov, Oleg, 2002. "Einflussfaktoren Auf Gewinn Und Rentabilität Landwirtschaftlicher Großbetriebe Gebiet Omsk, Russland," IAMO Discussion Papers 14919, Institute of Agricultural Development in Transition Economies (IAMO).
    17. Yao Wu & David Levinson, 2005. "The Rational Locator Reexamined," Working Papers 200503, University of Minnesota: Nexus Research Group.
    18. Kameng Nip & Zhenbo Wang & Zizhuo Wang, 2021. "Assortment Optimization under a Single Transition Choice Model," Production and Operations Management, Production and Operations Management Society, vol. 30(7), pages 2122-2142, July.
    19. B. S. C. Campello & C. T. L. S. Ghidini & A. O. C. Ayres & W. A. Oliveira, 2022. "A residual recombination heuristic for one-dimensional cutting stock problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 194-220, April.
    20. Koijen, R.S.J. & Nijman, T.E. & Werker, B.J.M., 2006. "Optimal Portfolio Choice with Annuitization," Discussion Paper 2006-78, Tilburg University, Center for Economic Research.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0426-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.