D'Onofrio model (binary variant)

D'Onofrio model (binary variant)
State dimension:	1
Differential states:	4
Discrete control functions:	4
Path constraints:	2

This site describes a D'Onofrio model variant with four binary controls instead which of only two continuous controls. The continuous controls are replaced via the outer convexifacation method.

Mathematical formulation

For $t \in [t_0, t_f]$ the optimal control problem is given by

$\begin{array}{llcl} \displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & - \zeta x_0 \text{ln} \left( \frac{x_0}{x_1} \right) - \sum\limits_{i=1}^{4} w_i\;c_{1,i}\; F \; x_0 , \\ & \dot{x}_1 & = & b x_0 - \mu x_1 - d x_0^{\frac{2}{3}}x_1 -\sum\limits_{i=1}^{4} w_i c_{0,i} \; G x_1 - \sum\limits_{i=1}^{4} w_i\;c_{1,i} \; \eta x_1, \\ & \dot{x}_2 & = & \sum\limits_{i=1}^{4} w_i\;c_{0,i}, \\ & \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex] & x_2 & \leq & x_2^{max}, \\ & x_3 & \leq & x_3^{max},\\ & 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\ & w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4. \end{array}$

Parameters

The parameters and scenarios are as in D'Onofrio_chemotherapy_model, the new fixed parameters are

$(c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0), \qquad (c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_1^{max},u_1^{max},0).$

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 scenario 2 of the relaxed problem with $n_t=6000, \, n_u=100$ is $19.3561387$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $169.45773$. The binary control solution was evaluated in the MIOCP by using a Merit function with additional Mayer term $100 \max\limits_{t\in[0,1]}\{0,x_2(t)-x_2^{max}\}+1000 \max\limits_{t\in[0,1]}\{0,x_3(t)-x_3^{max}\}$.

• Reference solution plots
• DonofriaRelaxed 6000 60 1.png

  Optimal relaxed differential states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=6000, \, n_u=100$.

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  Optimal binary differential states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=6000, \, n_u=100$. The relaxed controls were approximated by Combinatorial Integral Approximation.

Source Code

Model description is available in

• AMPL code at D'Onofrio model, binary variant (AMPL)