Egerstedt standard problem

Egerstedt standard problem
State dimension:	1
Differential states:	3
Discrete control functions:	3
Path constraints:	1
Interior point equalities:	3

The Egerstedt standard problemm is the problem is of an academic nature and was proposed by Egerestedt to highlight the features of an Hybrid System algorithm in 2006 [Egerstedt2006]Author: M. Egerstedt; Y. Wardi; H. Axelsson
Journal: IEEE Transactions on Automatic Control
Pages: 110--115
Title: Transition-time optimization for switched-mode dynamical systems
Volume: 51
Year: 2006
. It has been used since then in many MIOCP research studies (e.g. [Jung2013]Author: M. Jung; C. Kirches; S. Sager
Booktitle: Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel
Editor: M. J\"unger and G. Reinelt
Pages: 387--417
Publisher: Springer Berlin Heidelberg
Title: On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control
Url: http://www.mathopt.de/PUBLICATIONS/Jung2013.pdf
Year: 2013
) for benchmarking of MIOCP algorithms.

Mathematical formulation

The mixed-integer optimal control problem after partial outer convexification is given by

$\begin{array}{ll}\underset{x,\omega }{\min }& x_3(t_f)\\ \text{s.t.}& {\dot{x}}_1& =& -x_1{\omega }_1+(x_1+x_2){\omega }_2+(x_1-x_2){\omega }_3,\\ & {\dot{x}}_2& =& (x_1+2x_2){\omega }_1+(x_1-2x_2){\omega }_2+(x_1+x_2){\omega }_3,\\ & {\dot{x}}_3& =& x_1^2+x_2^2,\\ & x(0)& =& (0.5,0.5,0{)}^T,\\ & x_2(t)& \ge & 0.4,\\ & 1& =& \sum _{i=1}^3{\omega }_i(t),\\ & \omega (t)& \in & \{0,1\},\end{array}$

for $t\in [t_0,t_f]=[0,1]$.

Reference Solutions

If the problem is relaxed, i.e., we demand that $w(t)$ be in the continuous interval $[0,1]$ instead of the binary choice $\{0,1\}$, the optimal solution can be determined by using a direct method such as collocation or Bock's direct multiple shooting method.

The optimal objective value of the relaxed problem with $n_t=6000,n_u=40$ is $x_3(t_f)=1.0.995906234$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $x_3(t_f)=3.20831942$. The binary control solution was evaluated in the MIOCP by using a Merit function with additional Lagrange term $100\max {}_{t\in [0,1]}\{0,0.4-x_2(t)\}$.

• Reference solution plots
• 
  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=12000,n_u=400$.
• 
  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=12000,n_u=400$. The relaxed controls were approximated by Combinatorial Integral Approximation.

Source Code

Model descriptions are available in

• ACADO code at Lotka Volterra fishing problem (ACADO)
• AMPL code at Lotka Volterra fishing problem (AMPL)
• APMonitor code at Lotka Volterra fishing problem (APMonitor)
• Bocop code at Lotka Volterra fishing problem (Bocop)
• Casadi code at Lotka Volterra fishing problem (Casadi)
• JModelica code at Lotka Volterra fishing problem (JModelica)
• JuMP code at Lotka Volterra fishing problem (JuMP)
• Muscod code at Lotka Volterra fishing problem (Muscod)
• switch code at Lotka Volterra fishing problem (switch)
• PROPT code at Lotka Volterra fishing problem (TomDyn/PROPT)

Variants

There are several alternative formulations and variants of the above problem, in particular

• a prescribed time grid for the control function [Sager2006]Address: Heidelberg
  Author: S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder
  Booktitle: Recent Advances in Optimization
  Editor: A. Seeger
  Note: ISBN 978-3-5402-8257-0
  Pages: 269--289
  Publisher: Springer
  Series: Lectures Notes in Economics and Mathematical Systems
  Title: Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem
  Volume: 563
  Year: 2009
  , see also Lotka Volterra fishing problem (AMPL),
• a time-optimal formulation to get into a steady-state [Sager2005]Address: Tönning, Lübeck, Marburg
  Author: S. Sager
  Editor: ISBN 3-89959-416-9
  Publisher: Der andere Verlag
  Title: Numerical methods for mixed--integer optimal control problems
  Url: http://mathopt.de/PUBLICATIONS/Sager2005.pdf
  Year: 2005
  ,
• the usage of a different target steady-state, as the one corresponding to $w(t)=1$ which is $(1+c_1,1-c_0)$, see Lotka Volterra multi-arcs problem
• different fishing control functions for the two species, see Lotka Volterra Multimode fishing problem
• different parameters and start values.

Miscellaneous and Further Reading

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper [Sager2006]Address: Heidelberg
Author: S. Sager; H.G. Bock; M. Diehl; G. Reinelt; J.P. Schl\"oder
Booktitle: Recent Advances in Optimization
Editor: A. Seeger
Note: ISBN 978-3-5402-8257-0
Pages: 269--289
Publisher: Springer
Series: Lectures Notes in Economics and Mathematical Systems
Title: Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem
Volume: 563
Year: 2009
and revisited in his PhD thesis [Sager2005]Address: Tönning, Lübeck, Marburg
Author: S. Sager
Editor: ISBN 3-89959-416-9
Publisher: Der andere Verlag
Title: Numerical methods for mixed--integer optimal control problems
Url: http://mathopt.de/PUBLICATIONS/Sager2005.pdf
Year: 2005
. These are also the references to look for more details.

References

[Egerstedt2006]	M. Egerstedt; Y. Wardi; H. Axelsson (2006): Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51, 110--115
[Jung2013]	M. Jung; C. Kirches; S. Sager (2013): On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control. Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel

We present numerical results for a benchmark MIOCP from a previous study [157] with the addition of switching constraints. In its original form, the problem was:

After partial outer convexification with respect to the integer control v, the binary convexified counterpart problem reads