Lotka Volterra absolute fishing problem

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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.