Dynamic Optimization

plt.plot(solution1['time'],solution1['p'],'k-',linewidth=2)
plt.plot(solution2['time'],solution2['p'],'b--',linewidth=2)

plt.plot(solution1['time'],solution1['x'],'r--',linewidth=2)
plt.plot(solution2['time'],solution2['x'],'g:',linewidth=2)

The spread of HIV in a patient is approximated with balance equations on (H)ealthy, (I)nfected, and (V)irus population counts'^2^'.

from apm import *

from apm import *  # load APMonitor library

fid.write(' p[1:'+str(n)+'][1::'+str(m)+'] \n')

Attach:download.png [[Attach:apmonitor_matrix.zip|Initialize Parameter Matrix]]

The following example is a demonstration of inserting different initial conditions or parameter values at points throughout the time horizon. A simulation solution is used to provide guess values for a subsequent simulation. The parameter [[https://apmonitor.com/wiki/index.php/Main/OptionApmCsvRead|CSV_READ]] can be set to ''2'' to provide the initial values for a calculated state. The default (''CSV_READ=1'') only updates the fixed values and skips the values that are calculated by the solver.

Simulation is a first step after the model development to verify convergence, validate the model response to input changes, and manually adjust parameters to fit an expected response. This section demonstrates how to set up and initialize a dynamic model simulation. Options such as [[Main/OptionApmCsvRead|CSV_READ]] control how much information is read from a data (CSV) file.

The following example is a demonstration of inserting different initial conditions or parameter values at points throughout the time horizon. A simulation solution is used to provide guess values for a subsequent simulation. The parameter [[Main/OptionApmCsvRead|CSV_READ]] can be set to ''2'' to provide the initial values for a calculated state. The default (''CSV_READ=1'') only updates the fixed values and skips the values that are calculated by the solver.

Simulation is a first step after the model development to verify convergence, validate the model response to input changes, and manually adjust parameters to fit an expected response. This section demonstrates how to set up and initialize a dynamic model simulation. A first example shows how to use a scripting language such as MATLAB or Python to provide input values for parameters.

The following example is a demonstration of inserting different initial conditions or parameter values at points throughout the time horizon. A simulation solution is used to provide guess values for a subsequent simulation.

Simulation is a first step in after the model development to verify convergence, validate the model response to input changes, and manually adjust parameters to fit an expected response. This section demonstrates how to set up and solve a dynamic model.

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Although a problem may be written correctly, sometimes the solver fails to find a solution or requires excessive time to find a solution. Initialization strategies are critical in these situations to find a nearby solution that seeds the optimization problem with a starting point that allows convergence'^1^'.

# 5.Safdarnejad, S.M., Hedengren, J.D., Lewis, N.R., Haseltine, E., Initialization Strategies for Optimization of Dynamic Systems, Computers and Chemical Engineering, 2015, Vol. 78, pp. 39-50, DOI: 10.1016/j.compchemeng.2015.04.016. [[https://www.sciencedirect.com/science/article/pii/S0098135415001179 | Article]]

# Safdarnejad, S.M., Hedengren, J.D., Lewis, N.R., Haseltine, E., Initialization Strategies for Optimization of Dynamic Systems, Computers and Chemical Engineering, DOI: 10.1016/j.compchemeng.2015.04.016. [[https://www.sciencedirect.com/science/article/pii/S0098135415001179 | Article]]

With guess values for parameters (kr'_1..6_'), approximately match the laboratory data for this patient. A subsequent section introduces methods for parameter estimation by minimizing an objective function.

 dH/dt = kr^_1_^ - kr^_2_^ H - kr^_3_^ H V
 dI/dt = kr^_3_^ H V - kr^_4_^ I
 dV/dt = -kr^_3_^ H V - kr^_5_^ V + kr^_6_^ I
 LV = log^_10_^(V)

 kr^_1_^ = new healthy cells
 kr^_2_^ = death rate of healthy cells
 kr^_3_^ = healthy cells converting to infected cells
 kr^_4_^ = death rate of infected cells
 kr^_5_^ = death rate of virus
 kr^_6_^ = production of virus by infected cells

Although a problem may be written correctly, sometimes the solver fails to find a solution or requires excessive time to find a solution. Initialization strategies are critical in these situations to find a nearby solution that seeds the optimization problem with a starting point that allows convergence.

* Safdarnejad, S.M., Hedengren, J.D., Lewis, N.R., Haseltine, E., Initialization Strategies for Optimization of Dynamic Systems, Computers and Chemical Engineering, DOI: 10.1016/j.compchemeng.2015.04.016. [[https://www.sciencedirect.com/science/article/pii/S0098135415001179 | Article]]

Simulation is a first step in after the model development to verify convergence, validate the model response to input changes, and manually adjust parameters to fit an expected response. This section demonstrates how to set up and solve a simple dynamic model.

!!!! Solution

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