March  2020, 10(1): 39-43. doi: 10.3934/naco.2019031

Initial guess sensitivity in computational optimal control problems

1. 

Applied Mathematical Analysis, 24748 SE Mirromont Pl., Issaquah, WA 98027, USA

2. 

Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

* Corresponding author: Stephen L Campbell

Received  July 2018 Revised  April 2019 Published  May 2019

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.

Citation: John T. Betts, Stephen L. Campbell, Claire Digirolamo. Initial guess sensitivity in computational optimal control problems. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 39-43. doi: 10.3934/naco.2019031
References:
[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.

show all references

References:
[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.

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
$ i $ $ w_0 $ $ w_f $
1 1 0
2 -1 -1
3 1 -1
4 0 1
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
$ \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
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
$ \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
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
$ \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
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