American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations
  • NACO Home
  • This Issue
  • Next Article
    Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis
doi: 10.3934/naco.2019031

Initial guess sensitivity in Computational optimal control problems

John T. Betts 1, , Stephen L. Campbell 2,, and  Claire Digirolamo 2,

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.

Keywords: Optimal control, initial guesses, interior point methods, IPOPT, GPOPS II.
Mathematics Subject Classification: Primary: 49M37; Secondary: 49K40, 49M25.
Citation: John T. Betts, Stephen L. Campbell, Claire Digirolamo. Initial guess sensitivity in Computational optimal control problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019031
References:
[1]

J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010. Google Scholar

[2]

C. L. Darby, W. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010. Google Scholar

[2]

C. L. Darby, W. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
Table Options

Download as excel

$ \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
[1]

Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
[2]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[3]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092
[4]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[5]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[6]

Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023
[7]

Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the P*NCP. Journal of Industrial & Management Optimization, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569
[8]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[9]

Behrouz Kheirfam, Guoqiang Wang. An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 171-184. doi: 10.3934/naco.2017011
[10]

Liming Sun, Li-Zhi Liao. An interior point continuous path-following trajectory for linear programming. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1517-1534. doi: 10.3934/jimo.2018107
[11]

Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981
[12]

Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications. Inverse Problems & Imaging, 2007, 1 (3) : 507-523. doi: 10.3934/ipi.2007.1.507
[13]

Ming Mei, Yau Shu Wong, Liping Liu. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 839-857. doi: 10.3934/dcdsb.2007.7.839
[14]

Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211
[15]

Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601
[16]

Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949
[17]

Yanqin Bai, Pengfei Ma, Jing Zhang. A polynomial-time interior-point method for circular cone programming based on kernel functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 739-756. doi: 10.3934/jimo.2016.12.739
[18]

Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891
[19]

Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
[20]

Yinghong Xu, Lipu Zhang, Jing Zhang. A full-modified-Newton step infeasible interior-point algorithm for linear optimization. Journal of Industrial & Management Optimization, 2016, 12 (1) : 103-116. doi: 10.3934/jimo.2016.12.103

 Impact Factor: 

Tools

Metrics

  • PDF downloads (27)
  • HTML views (280)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences