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Up-> CppAD cppad_ipopt_nlp cppad_ipopt_ode ipopt_ode_simple

CppAD-> Install Introduction AD ADFun multi_thread library cppad_ipopt_nlp Example preprocessor Appendix

cppad_ipopt_nlp-> cppad_ipopt_windows ipopt_get_started.cpp cppad_ipopt_ode ipopt_ode_speed.cpp

cppad_ipopt_ode-> ipopt_ode_problem ipopt_ode_simple ipopt_ode_fast ipopt_ode_run.hpp ipopt_ode_check.cpp

ipopt_ode_simple-> ipopt_ode_simple.hpp

Headings-> Purpose Argument Vector Objective Function Initial Condition Constraint Trapezoidal Approximation Constraint Source

ODE Fitting Using Simple Representation
Purpose
In this section we represent the objective and constraint functions, (in the simultaneous forward and reverse optimization problem) using the simple representation in the sense of cppad_ipopt_nlp.

Argument Vector
The argument vector that we are optimizing with respect to ( $\mathit{x}$ in cppad_ipopt_nlp ) has the following structure $$\mathit{x}=({\mathit{y}}^0,\cdots ,{\mathit{y}}^{\mathit{S}(\mathit{Nz})},\mathit{a})$$ Note that $\mathit{x}\in {\mathbf{R}}^{\mathit{S}(\mathit{Nz})+\mathit{Na}}$ and $$\begin{array}{ccc}\hfill {\mathit{y}}^{\mathit{i}}& \hfill =\hfill & ({\mathit{x}}_{\mathit{Ny}*\mathit{i}},\dots ,{\mathit{x}}_{\mathit{Ny}*\mathit{i}+\mathit{Ny}-1})\hfill \\ \hfill \mathit{a}& \hfill =\hfill & ({\mathit{x}}_{\mathit{Ny}*\mathit{S}(\mathit{Nz})+\mathit{Ny}},\dots ,{\mathit{x}}_{\mathit{Ny}*\mathit{S}(\mathit{Nz})+\mathit{Na}-1})\hfill \end{array}$$
Objective Function
The objective function ( ${\mathit{fg}}_0(\mathit{x})$ in cppad_ipopt_nlp ) has the following representation, $${\mathit{fg}}_0(\mathit{x})=\sum _{\mathit{i}=1}^{\mathit{Nz}}{\mathit{H}}_{\mathit{i}}({\mathit{y}}^{\mathit{S}(\mathit{i})},\mathit{a})$$
Initial Condition Constraint
For $\mathit{i}=1,\dots ,\mathit{Ny}$ , we define the component functions ${\mathit{fg}}_{\mathit{i}}(\mathit{x})$ , and corresponding constraint equations, by $$0={\mathit{fg}}_{\mathit{i}}(\mathit{x})={\mathit{y}}_{\mathit{i}}^0-{\mathit{F}}_{\mathit{i}}(\mathit{a})$$
Trapezoidal Approximation Constraint
For $\mathit{i}=1,\dots ,\mathit{S}(\mathit{Nz})$ , and for $\mathit{j}=1,\dots ,\mathit{Ny}$ , we define the component functions ${\mathit{fg}}_{\mathit{Ny}*\mathit{i}+\mathit{j}}(\mathit{x})$ , and corresponding constraint equations, by $$0={\mathit{fg}}_{\mathit{Ny}*\mathit{i}+\mathit{j}}={\mathit{y}}_{\mathit{j}}^{\mathit{i}}-{\mathit{y}}_{\mathit{j}}^{\mathit{i}-1}-\left[{\mathit{G}}_{\mathit{j}}({\mathit{y}}^{\mathit{i}},\mathit{a})+{\mathit{G}}_{\mathit{j}}({\mathit{y}}^{\mathit{i}-1},\mathit{a})\right]*\frac{{\mathit{t}}_{\mathit{i}}-{\mathit{t}}_{\mathit{i}-1}}{2}$$
Source
The file ipopt_ode_simple.hpp contains source code for this representation of the objective and constraints.

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Input File: cppad_ipopt/example/ode2.omh