D'Onofrio model (binary variant): Difference between revisions

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

← Older edit

Latest revision as of 09:02, 2 February 2026

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

$\begin{array}{llcl} \min_{x, u} & x_{0} (t_{f}) & + & α \int_{t_{0}}^{t_{f}} u_{0} (t)^{2} d t \\ s.t. & {\dot{x}}_{0} & = & - ζ x_{0} ln (\frac{x_{0}}{x_{1}}) - \sum_{i = 1}^{4} w_{i} c_{1, i} F x_{0}, \\ {\dot{x}}_{1} & = & b x_{0} - μ x_{1} - d x_{0}^{\frac{2}{3}} x_{1} - \sum_{i = 1}^{4} w_{i} c_{0, i} G x_{1} - \sum_{i = 1}^{4} w_{i} c_{1, i} η x_{1}, \\ {\dot{x}}_{2} & = & \sum_{i = 1}^{4} w_{i} c_{0, i}, \\ {\dot{x}}_{3} & = & \sum_{i = 1}^{4} w_{i} c_{1, i}, \\ x_{2} & \leq & x_{2}^{m a x}, \\ x_{3} & \leq & x_{3}^{m a x}, \\ 1 & = & \sum_{i = 1}^{4} w_{i} (t), \\ w_{i} (t) & \in & {0, 1}, i = 1 \dots 4 . \end{array}$

Parameters

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

$(c_{0, 1}, c_{0, 2}, c_{0, 3}, c_{0, 4}) = (u_{0}^{m a x}, u_{0}^{m a x}, 0, 0), (c_{1, 1}, c_{1, 2}, c_{1, 3}, c_{1, 4}) = (0, u_{1}^{m a x}, u_{1}^{m a x}, 0) .$

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 $100 \max_{t \in [0, 1]} {0, x_{2} (t) - x_{2}^{m a x}} + 1000 \max_{t \in [0, 1]} {0, x_{3} (t) - x_{3}^{m a x}}$ .

Reference solution plots
Optimal relaxed differential states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 6000, n_{u} = 100$ .
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)

@@ Line 17: / Line 17: @@
   \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}_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]
+              & \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}^{3}w_i(t), \\
+& 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\
   & w_i(t) &\in&  \{0, 1\}, \quad i=1\ldots 4.
 \end{array}
 </math>
 </p>
 == Parameters ==
-These fixed values are used within the model:
+The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are
-<math>[t_0,t_f]=[0,20], c_1=-1, c_2=0.75, c_3=-2.</math>
+<math>(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).
+</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 direct optimal control.
+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  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  <math> n_t=6000, \, n_u=100  </math> is <math>19.3561387</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>169.45773</math>.  The binary control solution was evaluated in the MIOCP by using a Merit function with additional Mayer term
+<math> 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}\}  </math>.
-The optimal objective value of the relaxed problem with  <math> n_t=6000, \, n_u=60  </math> is <math>1.30167235</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>1.30273681</math>.
 <gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
-  Image:VanderpolRelaxed 6000 100 1.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=60</math>.
+  Image:DonofrioRelaxed 6000 60 1.png| Optimal relaxed differential states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=100</math>.
-  Image:VanderpolCIA_6000_100_1.pdf| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
+  Image:DonofrioCIA 6000 60 1.png| Optimal binary differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
 </gallery>
 == Source Code ==
 Model description is available in
-* [[:Category:AMPL | AMPL code]] at [[Van der Pol Oscillator binary variant(AMPL)]]
+* [[:Category:AMPL | AMPL code]] at [[D'Onofrio model, binary variant (AMPL)]]