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

$\begin{array}{llcl}\underset{x,u}{\min }& x_0(t_f)& +& \alpha \underset{t_0}{\overset{t_f}{\int }}{u}_0(t{)}^2\text{d}t\\ \text{s.t.}& {\dot{x}}_0& =& -\zeta x_0\text{ln}\left(\frac{x_0}{x_1}\right)-\sum _{i=1}^4{w}_i{c}_{1,i}F{x}_0,\\ & {\dot{x}}_1& =& b{x}_0-\mu x_1-d{x}_0^{\frac{2}{3}}{x}_1-\sum _{i=1}^4{w}_i{c}_{0,i}G{x}_1-\sum _{i=1}^4{w}_i{c}_{1,i}\eta x_1,\\ & {\dot{x}}_2& =& \sum _{i=1}^4{w}_i{c}_{0,i},\\ & {\dot{x}}_3& =& \sum _{i=1}^4{w}_i{c}_{1,i},\\ [1.5ex]& x_2& \le & x_2^{max},\\ & x_3& \le & x_3^{max},\\ & 1& =& \sum _{i=1}^4{w}_i(t),\\ & w_i(t)& \in & \{0,1\},i=1\dots 4.\end{array}$

Parameters

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

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

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 $100\max {}_{t\in [0,1]}\{0,x_2(t)-x_2^{max}\}+1000\max {}_{t\in [0,1]}\{0,x_3(t)-x_3^{max}\}$.

• Reference solution plots
• 
  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=6000,n_u=60$.
• 
  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=6000,n_u=60$. 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)