Initial guess sensitivity in computational optimal control problems

John T. Betts; Stephen L. Campbell; Claire Digirolamo

doi:10.3934/naco.2019031

Article Contents

2020, Volume 10, Issue 1: 39-43. Doi: 10.3934/naco.2019031

This issue Previous Article Numerical solution of an obstacle problem with interval coefficients Next Article Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model

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
^* Corresponding author: Stephen L Campbell

Received: July 2018

Revised: April 2019

Published: March 2020

Abstract Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by

Abstract

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:

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

Citation:

FullText(HTML)

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

Download: Full-size image 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

| Show Table

DownLoad: CSV

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

Related Papers

Cited by

References

[1]	J. T. Betts, Methods for Optimal Control and Estimation using Nonlinear Programming, SIAM, Philadelphia, 2010.
[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.
[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.