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

$\begin{array}{llcl} \min_{x, u} & x_{0} (t_{f}) & + & α \int_{t_{0}}^{t_{f}} u_{0} (t)^{2} d t \\ s.t. & {\dot{x}}_{0} & = & - ζ x_{0} ln (\frac{x_{0}}{x_{1}}) - \sum_{i = 1}^{4} w_{i} c_{1, i} F x_{0}, \\ {\dot{x}}_{1} & = & b x_{0} - μ x_{1} - d x_{0}^{\frac{2}{3}} x_{1} - \sum_{i = 1}^{4} w_{i} c_{0, i} G x_{1} - \sum_{i = 1}^{4} w_{i} c_{1, i} η x_{1}, \\ {\dot{x}}_{2} & = & \sum_{i = 1}^{4} w_{i} c_{0, i}, \\ {\dot{x}}_{3} & = & \sum_{i = 1}^{4} w_{i} c_{1, i}, \\ [1.5 e x] & x_{2} & \leq & x_{2}^{m a x}, \\ x_{3} & \leq & x_{3}^{m a x}, \\ 1 & = & \sum_{i = 1}^{4} w_{i} (t), \\ w_{i} (t) & \in & {0, 1}, i = 1 \dots 4 . \end{array}$

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)