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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
Use Ipopt to Solve a Nonlinear Programming Problem

Syntax
# include <cppad/ipopt/solve.hpp>
 ipopt::solve(
    options, xi, xl, xu, gl, gu, fg_eval, solution
)

Purpose
The function ipopt::solve solves nonlinear programming problems of the form @[@ \begin{array}{rll} {\rm minimize} & f (x) \\ {\rm subject \; to} & gl \leq g(x) \leq gu \\ & xl \leq x \leq xu \end{array} @]@ This is done using Ipopt optimizer and CppAD for the derivative and sparsity calculations.

Include File
If include_ipopt is on the cmake command line, the file cppad/ipopt/solve.hpp is included by cppad/cppad.hpp. Otherwise, cppad/ipopt/solve.hpp can be included directly (If cppad/cppad.hpp has not yet been included, cppad/ipopt/solve.hpp will automatically include it.)

Bvector
The type Bvector must be a SimpleVector class with elements of type bool.

Dvector
The type DVector must be a SimpleVector class with elements of type double.

options
The argument options has prototype
    const std::string options
 It contains a list of options. Each option, including the last option, is terminated by the '\n' character. Each line consists of two or three tokens separated by one or more spaces.

Retape
You can set the retape flag with the following syntax:
    Retape value
 If the value is true, ipopt::solve with retape the operation sequence for each new value of x . If the value is false, ipopt::solve will tape the operation sequence at the value of xi and use that sequence for the entire optimization process. The default value is false.

Sparse
You can set the sparse Jacobian and Hessian flag with the following syntax:
    Sparse value direction
 If the value is true, ipopt::solve will use a sparse matrix representation for the computation of Jacobians and Hessians. Otherwise, it will use a full matrix representation for these calculations. The default for value is false. If sparse is true, retape must be false.

It is unclear if sparse_jacobian would be faster user forward or reverse mode so you are able to choose the direction. If
    value == true && direction == forward
 the Jacobians will be calculated using SparseJacobianForward. If
    value == true && direction == reverse
 the Jacobians will be calculated using SparseJacobianReverse.

String
You can set any Ipopt string option using a line with the following syntax:
    String name value
 Here name is any valid Ipopt string option and value is its setting.

Numeric
You can set any Ipopt numeric option using a line with the following syntax:
    Numeric name value
 Here name is any valid Ipopt numeric option and value is its setting.

Integer
You can set any Ipopt integer option using a line with the following syntax:
    Integer name value
 Here name is any valid Ipopt integer option and value is its setting.

xi
The argument xi has prototype
    const Vector& xi
 and its size is equal to nx . It specifies the initial point where Ipopt starts the optimization process.

xl
The argument xl has prototype
    const Vector& xl
 and its size is equal to nx . It specifies the lower limits for the argument in the optimization problem.

xu
The argument xu has prototype
    const Vector& xu
 and its size is equal to nx . It specifies the upper limits for the argument in the optimization problem.

gl
The argument gl has prototype
    const Vector& gl
 and its size is equal to ng . It specifies the lower limits for the constraints in the optimization problem.

gu
The argument gu has prototype
    const Vector& gu
 and its size is equal to ng . It specifies the upper limits for the constraints in the optimization problem.

fg_eval
The argument fg_eval has prototype
    FG_eval fg_eval
 where the class FG_eval is unspecified except for the fact that it supports the syntax
    FG_eval::ADvector
    fg_eval(fg, x)
 The type ADvector and the arguments to fg , x have the following meaning:

ADvector
The type FG_eval::ADvector must be a SimpleVector class with elements of type AD<double>.

x
The fg_eval argument x has prototype
    const ADvector& x
 where nx = x.size() .

fg
The fg_eval argument fg has prototype
    ADvector& fg
 where 1 + ng = fg.size() . The input value of the elements of fg does not matter. Upon return from fg_eval ,
    fg[0] = @(@ f (x) @)@
and for @(@ i = 0, \ldots , ng-1 @)@,
    fg[1 + i] = @(@ g_i (x) @)@

solution
The argument solution has prototype
    ipopt::solve_result<Dvector>& solution
 After the optimization process is completed, solution contains the following information:

status
The status field of solution has prototype
    ipopt::solve_result<Dvector>::status_type solution.status
 It is the final Ipopt status for the optimizer. Here is a list of the possible values for the status:
status Meaning
not_defined The optimizer did not return a final status for this problem.
unknown The status returned by the optimizer is not defined in the Ipopt documentation for finalize_solution.
success Algorithm terminated successfully at a point satisfying the convergence tolerances (see Ipopt options).
maxiter_exceeded The maximum number of iterations was exceeded (see Ipopt options).
stop_at_tiny_step Algorithm terminated because progress was very slow.
stop_at_acceptable_point Algorithm stopped at a point that was converged, not to the 'desired' tolerances, but to 'acceptable' tolerances (see Ipopt options).
local_infeasibility Algorithm converged to a non-feasible point (problem may have no solution).
user_requested_stop This return value should not happen.
diverging_iterates It the iterates are diverging.
restoration_failure Restoration phase failed, algorithm doesn't know how to proceed.
error_in_step_computation An unrecoverable error occurred while Ipopt tried to compute the search direction.
invalid_number_detected Algorithm received an invalid number (such as nan or inf) from the users function fg_info.eval or from the CppAD evaluations of its derivatives (see the Ipopt option check_derivatives_for_naninf).
internal_error An unknown Ipopt internal error occurred. Contact the Ipopt authors through the mailing list.

x
The x field of solution has prototype
    Vector solution.x
 and its size is equal to nx . It is the final @(@ x @)@ value for the optimizer.

zl
The zl field of solution has prototype
    Vector solution.zl
 and its size is equal to nx . It is the final Lagrange multipliers for the lower bounds on @(@ x @)@.

zu
The zu field of solution has prototype
    Vector solution.zu
 and its size is equal to nx . It is the final Lagrange multipliers for the upper bounds on @(@ x @)@.

g
The g field of solution has prototype
    Vector solution.g
 and its size is equal to ng . It is the final value for the constraint function @(@ g(x) @)@.

lambda
The lambda field of solution has prototype
    Vector> solution.lambda
 and its size is equal to ng . It is the final value for the Lagrange multipliers corresponding to the constraint function.

obj_value
The obj_value field of solution has prototype
    double solution.obj_value
 It is the final value of the objective function @(@ f(x) @)@.

Example
All the examples return true if it succeeds and false otherwise.

get_started
The file example/ipopt_solve/get_started.cpp is an example and test of ipopt::solve taken from the Ipopt manual.

retape
The file example/ipopt_solve/retape.cpp demonstrates when it is necessary to specify retape as true.

ode_inverse
The file example/ipopt_solve/ode_inverse.cpp demonstrates using Ipopt to solve for parameters in an ODE model.
Input File: include/cppad/ipopt/solve.hpp