D'Onofrio model (binary variant)

From mintOC
Jump to: navigation, search
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

 \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.


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

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}\}  .

Source Code

Model description is available in