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

Author

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    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.
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    Cited by:

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    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.

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