D'Onofrio model (binary variant) - Revision history

RobertLampel: /* Mathematical formulation */

2026-02-02T09:02:37Z

Mathematical formulation

← Older revision		Revision as of 09:02, 2 February 2026
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	& \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}_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_2 & \leq & x_2^{max}, \\
	& x_3 & \leq & x_3^{max},\\		& x_3 & \leq & x_3^{max},\\
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	</math>		</math>
	</p>		</p>


	== Parameters ==		== Parameters ==

ClemensZeile: /* Reference Solutions */

2018-01-11T15:33:18Z

Reference Solutions

← Older revision		Revision as of 15:33, 11 January 2018
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~DonofriaRelaxed~~ 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: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: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.		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>		</gallery>



	== Source Code ==		== Source Code ==

ClemensZeile at 15:32, 11 January 2018

2018-01-11T15:32:57Z

← Older revision		Revision as of 15:32, 11 January 2018
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<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:DonofriaRelaxed 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>		</gallery>

ClemensZeile at 14:38, 11 January 2018

2018-01-11T14:38:08Z

← Older revision		Revision as of 14:38, 11 January 2018
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	The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are		The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are

	<math>(c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0),		<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_0~~^{max},~~u_0~~^{max},0).		(c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_1^{max},u_1^{max},0).
	</math>		</math>

	== Reference Solutions ==		== 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">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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	Model description is available in		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)]]

ClemensZeile at 14:25, 11 January 2018

2018-01-11T14:25:17Z

← Older revision		Revision as of 14:25, 11 January 2018
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	The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are		The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are

	<math>(c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0) \\		<math>(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_0^{max},u_0^{max},0).		(c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_0^{max},u_0^{max},0).
	</math>		</math>

ClemensZeile at 14:24, 11 January 2018

2018-01-11T14:24:33Z

← Older revision		Revision as of 14:24, 11 January 2018
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	\displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex]		\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 , \\		\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}_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_2 & \leq & x_2^{max}, \\
	& x_3 & \leq & x_3^{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.		& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4.
	\end{array}		\end{array}
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	== Parameters ==		== 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) \\
			(c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_0^{max},u_0^{max},0).
			</math>

	== Reference Solutions ==		== Reference Solutions ==

ClemensZeile at 14:19, 11 January 2018

2018-01-11T14:19:31Z

← Older revision		Revision as of 14:19, 11 January 2018
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	& \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_2 & \leq & x_2^{max}, \\
	& x_3 & \leq & x_3^{max},		& x_3 & \leq & x_3^{max},\\
	& 1 &=& \sum\limits_{i=1}^{3}w_i(t), \\		& 1 &=& \sum\limits_{i=1}^{3}w_i(t), \\
	& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4.		& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4.

ClemensZeile: Created page with "{{Dimensions |nd = 1 |nx = 4 |nw = 4 |nc = 2 }}This site describ..."

2018-01-11T14:18:05Z

Created page with "{{Dimensions |nd = 1 |nx = 4 |nw = 4 |nc = 2 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 4
|nw = 4
|nc = 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 <math>t \in [t_0, t_f]</math> the optimal control problem is given by

<p>
<math>
\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}^{3}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:

<math>[t_0,t_f]=[0,20], c_1=-1, c_2=0.75, c_3=-2.</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.

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: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.
</gallery>

== Source Code ==

Model description is available in
* [[:Category:AMPL | AMPL code]] at [[Van der Pol Oscillator binary variant(AMPL)]]


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]