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Initial guess sensitivity in computational optimal control problems

  • * Corresponding author: Stephen L Campbell

    * Corresponding author: Stephen L Campbell 
Abstract Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by
  • An optimal control problem is presented that exhibited unexpected initial guess dependence when being solved with direct transcription methods. This note presents that example and the cautionary tale it provides.

    Mathematics Subject Classification: Primary: 49M37; Secondary: 49K40, 49M25.

    Citation:

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  • Figure 1.  Graphs of computed optimal control $ w $ for Example (2) with DT approach and $ M = 2, T = 1. $

    Table 1.  Endpoints for Initial $ w $ guesses $ w_i $ for Example 2

    $ i $ $ w_0 $ $ w_f $
    1 1 0
    2 -1 -1
    3 1 -1
    4 0 1
     | Show Table
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    Table 2.  Adjusted Values of $ \eta(T) $ for Example (2) using DT and $ M = 2 $, $ T = 1 $ and 3 iterations for different values of $ \gamma^2 $ and $ w_i $

    $ \gamma^2 $ $ w_1 $ $ w_2 $ $ w_3 $ $ w_4 $
    4.3 18.0613 5.3457 18.0613 18.0613
    4.4 18.0613 5.3457 18.0613 5.3457
    4.5 18.0613 18.0613 5.3457 5.3457
    4.6 18.0613 18.0613 18.0613 18.0613
    4.7 5.3457 5.3457 18.0613 5.3457
     | Show Table
    DownLoad: CSV

    Table 3.  Rerun of the results reported in Table Table 2. Bold entries have changed from Table 2

    $ \gamma^2 $ $ w_1 $ $ w_2 $ $ w_3 $ $ w_4 $
    4.3 18.0613 5.3457 18.0613 18.0613
    4.4 18.0613 5.3457 18.0613 5.3457
    4.5 18.0613 18.0613 5.3457 5.3457
    4.6 5.3457 18.0613 5.3457 18.0613
    4.7 5.3457 5.3457 18.0613 5.3457
     | Show Table
    DownLoad: CSV

    Table 4.  Adjusted Values of $ \eta(T) $ for Example (2) using DT and $ M = 2 $, $ T = 1 $ and 3 iterations for cost (3)

    $ \gamma^2 $ $ w_1 $ $ w_2 $ $ w_3 $ $ w_4 $
    4.3 18.0613 18.0613 5.3457 18.0613
    4.4 18.0613 18.0613 5.3457 5.3457
    4.5 18.0613 18.0613 5.3457 18.0613
    4.6 18.0613 18.0613 5.3457 18.0613
    4.7 18.0613 18.0613 5.3457 5.3457
     | Show Table
    DownLoad: CSV
  • [1] J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010.
    [2] C. L. DarbyW. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502.  doi: 10.1002/oca.957.
    [3] A. Forsgren, On warm starts for interior methods, in System Modeling and Optimization, IFIP International Federation for Information Processing (eds. F. Ceragioli, A. Dontchev, H. Furuta, K. Marti, and L. P. Pandolfi), Springer, Boston, 199 (2006), 51–66. doi: 10.1007/0-387-33006-2_6.
    [4] M. A. Patterson and A. V. Rao, GPOPS II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Transactions Mathematical Software, 41 (2014), 1-37.  doi: 10.1145/2558904.
    [5] A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: Gpops, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Transactions Mathematical Software, 37 (2010), 22: 1–22: 39. doi: 10.1145/2558904.
    [6] A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.
    [7] A. Wächter and L. T. Biegler, Failure of global convergence for a class of interior point methods for nonlinear programming, Mathematical Programming, 88 (2000), 565-574.  doi: 10.1007/PL00011386.
    [8] E. A. Yildirim and S. J. Wright, Warm start strategies in interior-point methods for linear programming, SIAM J. Optimization, 12 (2002), 782-810.  doi: 10.1137/S1052623400369235.
    [9] E. A. Yildirim, Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimensions, Computational Optimization and Applications, 41 (2008), 151-183.  doi: 10.1007/s10589-007-9096-y.
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