Lotka Volterra Multimode fishing problem - Revision history

ClemensZeile at 12:16, 21 December 2017

2017-12-21T12:16:27Z

← Older revision		Revision as of 12:16, 21 December 2017
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	== Reference Solutions ==		== Reference Solutions ==

	If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of ~~[http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle Pontryagins maximum principle]. The~~ optimal solution contains a singular arc, as can be seen in the plot of the optimal control. The two differential states and corresponding adjoint variables in the indirect approach are also displayed. A different approach to solving the relaxed problem is by using a direct method such as collocation or Bock's direct multiple shooting method. Optimal solutions for different control ~~discretizations are also plotted in the leftmost figure~~.		If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

	The optimal objective value of ~~this~~ relaxed problem is <math>~~x_2(t_f)~~ = ~~1.34408~~</math>~~. As follows from MIOC theory <bib id="Sager2011d" /> this~~ is the best lower bound on the optimal value of the original problem with the integer restriction on the control function. In other words, this objective value can be approximated arbitrarily close, if the control only switches often enough between 0 and 1. As no optimal solution exists, two suboptimal ones are shown, one with only two switches and an objective function value of <math>x_2(t_f) = 1.~~38276~~</math>~~, and one with 56 switches and~~ <math>x_2(t_f) = 1.~~34416~~</math>.		The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~lotkaRelaxedControls~~.png\| Optimal relaxed ~~control~~ determined by an ~~indirect approach and by a~~ direct approach ~~on different control discretization grids.~~		Image:MmlotkaRelaxed_12000_30_1.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=400</math>.
	~~Image:lotkaindirektStates.png\| Differential states~~ and ~~corresponding adjoint variables in the indirect approach~~.		Image:MmlotkaCIA 12000 30 1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	Image:~~lotka2Switches~~.png\| ~~Control~~ and ~~differential~~ states with ~~only two switches~~.
	~~Image:lotka56Switches.png\| Control and differential states with 56 switches~~.
	</gallery>		</gallery>

ClemensZeile at 09:43, 21 December 2017

2017-12-21T09:43:21Z

← Older revision		Revision as of 09:43, 21 December 2017
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	\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]		\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]
	\mbox{s.t.}		\mbox{s.t.}
	& \dot{x}_0 & = & x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{0,i} x_0 \; w_i, \\		& \dot{x}_0 & = & x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{0,i}\; x_0 \; w_i, \\
	& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{1,i} x_1 \; w_i, \\		& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{1,i}\; x_1 \; w_i, \\
	& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]		& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
	& x(0) &=& (0.5, 0.7, 0)^T, \\		& x(0) &=& (0.5, 0.7, 0)^T, \\
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	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 12],\\		[t_0, t_f] &=& [0, 12],\\
	(c_{0,1}, c_{1,1} &=& (0.2, 0.1),\\		(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
	(c_{0,2}, c_{1,2} &=& (0.4, 0.2),\\		(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
	(c_{0,3}, c_{1,3} &=& (0.01, 0.1).		(c_{0,3}, c_{1,3}) &=& (0.01, 0.1).
	\end{array}		\end{array}
	</math>		</math>
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	~~== Source Code ==~~

	~~Model descriptions are available in~~

	* [[:Category:ACADO \| ACADO code]] at [[Lotka Volterra fishing problem (ACADO)]]
	* [[:Category:AMPL \| AMPL code]] at [[Lotka Volterra fishing problem (AMPL)]]
	* [[:Category:APMonitor \| APMonitor code]] at [[Lotka Volterra fishing problem (APMonitor)]]
	* [[:Category:Bocop \| Bocop code]] at [[Lotka Volterra fishing problem (Bocop)]]
	* [[:Category:Casadi \| Casadi code]] at [[Lotka Volterra fishing problem (Casadi)]]
	* [[:Category:JModelica \| JModelica code]] at [[Lotka Volterra fishing problem (JModelica)]]
	* [[:Category:Julia/JuMP \| JuMP code]] at [[Lotka Volterra fishing problem (JuMP)]]
	* [[:Category:Muscod \| Muscod code]] at [[Lotka Volterra fishing problem (Muscod)]]
	* [[:Category:switch \| switch code]] at [[Lotka Volterra fishing problem (switch)]]
	* [[:Category:TomDyn/PROPT \| 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 <bib id="Sager2006" />, see also [[Lotka Volterra fishing problem (AMPL)]],
	* a time-optimal formulation to get into a steady-state <bib id="Sager2005" />,
	* the usage of a different target steady-state, as the one corresponding to <math> w(t) = 1</math> which is <math>(1 + c_1, 1 - c_0)</math>,
	* different fishing control functions for the two species,
	* different parameters and start values.

	~~== Miscellaneous and Further Reading ==~~
	The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details.

	~~== References ==~~
	~~<biblist />~~

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

ClemensZeile at 09:27, 20 December 2017

2017-12-20T09:27:02Z

← Older revision		Revision as of 09:27, 20 December 2017
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	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 12],\\		[t_0, t_f] &=& [0, 12],\\
	(c_{0,1}, c_{1,1) &=& (0.2, 0.1),\\		(c_{0,1}, c_{1,1} &=& (0.2, 0.1),\\
	(c_{0,2}, c_{1,2) &=& (0.4, 0.2),\\		(c_{0,2}, c_{1,2} &=& (0.4, 0.2),\\
	(c_{0,3}, c_{1,3) &=& (0.01, 0.1).		(c_{0,3}, c_{1,3} &=& (0.01, 0.1).
	\end{array}		\end{array}
	</math>		</math>

ClemensZeile at 09:26, 20 December 2017

2017-12-20T09:26:30Z

← Older revision		Revision as of 09:26, 20 December 2017
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	& x(0) &=& (0.5, 0.7, 0)^T, \\		& x(0) &=& (0.5, 0.7, 0)^T, \\
	& \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\		& \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\
	& w_i(t) &\in& \{0, 1\}.		& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 3.
	\end{array}		\end{array}
	</math>		</math>
	</p>		</p>

	Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. ~~The decision, whether the~~ fishing ~~fleet is actually fishing at time <math>t</math> is denoted by <math>w(t)</math>~~.		Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.

	== Parameters ==		== Parameters ==
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	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 12],\\		[t_0, t_f] &=& [0, 12],\\
	(~~c_0~~, ~~c_1~~) &=& (0.4, 0.2).		(c_{0,1}, c_{1,1) &=& (0.2, 0.1),\\
			(c_{0,2}, c_{1,2) &=& (0.4, 0.2),\\
			(c_{0,3}, c_{1,3) &=& (0.01, 0.1).
	\end{array}		\end{array}
	</math>		</math>

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2017-12-20T09:15:02Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 3 |nre = 3 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 3
|nre = 3
}}This site describes a Lotka Volterra variant with three binary controls instead of only one control.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llclr}
\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]
\mbox{s.t.}
& \dot{x}_0 & = & x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{0,i} x_0 \; w_i, \\
& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{1,i} x_1 \; w_i, \\
& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
& x(0) &=& (0.5, 0.7, 0)^T, \\
& \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\
& w_i(t) &\in& \{0, 1\}.
\end{array}
</math>
</p>

Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. The decision, whether the fishing fleet is actually fishing at time <math>t</math> is denoted by <math>w(t)</math>.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 12],\\
(c_0, c_1) &=& (0.4, 0.2).
\end{array}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of [http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle Pontryagins maximum principle]. The optimal solution contains a singular arc, as can be seen in the plot of the optimal control. The two differential states and corresponding adjoint variables in the indirect approach are also displayed. A different approach to solving the relaxed problem is by using a direct method such as collocation or Bock's direct multiple shooting method. Optimal solutions for different control discretizations are also plotted in the leftmost figure.

The optimal objective value of this relaxed problem is <math>x_2(t_f) = 1.34408</math>. As follows from MIOC theory <bib id="Sager2011d" /> this is the best lower bound on the optimal value of the original problem with the integer restriction on the control function. In other words, this objective value can be approximated arbitrarily close, if the control only switches often enough between 0 and 1. As no optimal solution exists, two suboptimal ones are shown, one with only two switches and an objective function value of <math>x_2(t_f) = 1.38276</math>, and one with 56 switches and <math>x_2(t_f) = 1.34416</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:lotkaRelaxedControls.png| Optimal relaxed control determined by an indirect approach and by a direct approach on different control discretization grids.
Image:lotkaindirektStates.png| Differential states and corresponding adjoint variables in the indirect approach.
Image:lotka2Switches.png| Control and differential states with only two switches.
Image:lotka56Switches.png| Control and differential states with 56 switches.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:ACADO | ACADO code]] at [[Lotka Volterra fishing problem (ACADO)]]
* [[:Category:AMPL | AMPL code]] at [[Lotka Volterra fishing problem (AMPL)]]
* [[:Category:APMonitor | APMonitor code]] at [[Lotka Volterra fishing problem (APMonitor)]]
* [[:Category:Bocop | Bocop code]] at [[Lotka Volterra fishing problem (Bocop)]]
* [[:Category:Casadi | Casadi code]] at [[Lotka Volterra fishing problem (Casadi)]]
* [[:Category:JModelica | JModelica code]] at [[Lotka Volterra fishing problem (JModelica)]]
* [[:Category:Julia/JuMP | JuMP code]] at [[Lotka Volterra fishing problem (JuMP)]]
* [[:Category:Muscod | Muscod code]] at [[Lotka Volterra fishing problem (Muscod)]]
* [[:Category:switch | switch code]] at [[Lotka Volterra fishing problem (switch)]]
* [[:Category:TomDyn/PROPT | 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 <bib id="Sager2006" />, see also [[Lotka Volterra fishing problem (AMPL)]],
* a time-optimal formulation to get into a steady-state <bib id="Sager2005" />,
* the usage of a different target steady-state, as the one corresponding to <math> w(t) = 1</math> which is <math>(1 + c_1, 1 - c_0)</math>,
* different fishing control functions for the two species,
* different parameters and start values.

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]