Difference between revisions of "Lotka Volterra absolute fishing problem"

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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.
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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 five different fishing options.
  
 
== Parameters ==
 
== Parameters ==
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== Reference Solutions ==
 
== Reference Solutions ==
  
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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.  
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If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control.  
  
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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>.   
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The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=150 </math> is <math>x_2(t_f) =0.345768563</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =0.348617982</math>.   
  
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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  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>.
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  Image:Lotka_abs_fish_relaxed_12000_80.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=150</math>.
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  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.
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  Image:Lotka_abs_fish_CIA_states_12000_80.pdf| Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
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Image:Lotka_abs_fish_CIA_controls_12000_80.pdf| Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.
 
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Latest revision as of 12:36, 14 October 2019

Lotka Volterra absolute fishing problem
State dimension: 1
Differential states: 3
Discrete control functions: 5
Interior point equalities: 3

This site describes a Lotka Volterra variant with five binary controls that all represent fishing of an absolute biomass.

Mathematical formulation

The mixed-integer optimal control problem is given by

\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}^{5} c_{0,i}\; w_i, \\ & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{1,i}\; 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}^{5}w_i(t) &=& 1, \\ & w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 5. \end{array}

Here the differential states (x_0, x_1) 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 \min \; x_2(t_f). This problem variant allows to choose between five different fishing options.

Parameters

These fixed values are used within the model.

\begin{array}{rcl} [t_0, t_f] &=& [0, 12],\\ (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),\\ (c_{0,4}, c_{1,4}) &=& (0, 0),\\ (c_{0,5}, c_{1,5}) &=& (-0.1, -0.2). \end{array}

Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) is in the continuous interval [0, 1] rather than being binary, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with n_t=12000, \, n_u=150 is x_2(t_f) =0.345768563. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is x_2(t_f) =0.348617982.

  • Reference solution plots

  • Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=12000, \, n_u=150.

  • Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=12000, \, n_u=150. The relaxed controls were approximated by Combinatorial Integral Approximation.

  • Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=12000, \, n_u=150. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.

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