/**
\page INTERFACES Interfacing your NLP to %Ipopt

\tableofcontents

\Ipopt has been designed to be flexible for a wide variety
of applications, and there are a number of ways to interface with
\Ipopt that allow specific data structures and linear
solver techniques. Nevertheless, the authors have included a standard
representation that should meet the needs of most users.

This tutorial will discuss six interfaces to \Ipopt, namely
the AMPL modeling language \cite FouGayKer:AMPLbook interface, and the C++,
C, Fortran, Java, and R code interfaces. AMPL is a modeling
language tool that allows users to write their optimization problem in a
syntax that resembles the way the problem would be written
mathematically. Once the problem has been formulated in AMPL, the
problem can be easily solved using the (already compiled)
\Ipopt AMPL solver executable, ipopt. Interfacing your
problem by directly linking code requires more effort to write, but can
be far more efficient for large problems.

We will illustrate how to use each of the four interfaces using an
example problem, number 71 from the Hock-Schittkowsky test suite \cite HS,
\f{equation}{ \begin{aligned}
    \min_{x \in \Re^4} &&x_1 x_4 (x_1 + x_2 + x_3)  +  x_3  \\
    \mbox{s.t.}  &&x_1 x_2 x_3 x_4 \ge 25  \\
    &&x_1^2 + x_2^2 + x_3^2 + x_4^2  =  40  \\
    &&1 \leq x_1, x_2, x_3, x_4 \leq 5, 
\end{aligned} \tag{HS071} \f} with the starting point
\f[ x_0 = (1, 5, 5, 1) \f] and the optimal solution
\f[ x_* = (1.00000000, 4.74299963, 3.82114998, 1.37940829). \f]

You can find further, less documented examples for using
\Ipopt from your own source code in the Ipopt/examples
subdirectory.

\section INTERFACE_AMPL Using Ipopt through AMPL

Using the AMPL solver executable is by far the easiest way to solve a
problem with \Ipopt. The user must simply formulate the
problem in AMPL syntax, and solve the problem through the AMPL
environment. There are drawbacks, however. AMPL is a 3rd party package
and, as such, must be appropriately licensed (a free student version for
limited problem size is available from the [AMPL website](http://www.ampl.com).
Furthermore, the AMPL environment may be
prohibitive for very large problems. Nevertheless, formulating the
problem in AMPL is straightforward and even for large problems, it is
often used as a prototyping tool before using one of the code
interfaces.

This tutorial is not intended as a guide to formulating models in AMPL.
If you are not already familiar with AMPL, please consult
\cite FouGayKer:AMPLbook.

The problem presented in (HS071) can be solved with \Ipopt with
the following AMPL model.

    # tell ampl to use the ipopt executable as a solver
    # make sure ipopt is in the path!
    option solver ipopt;

    # declare the variables and their bounds, 
    # set notation could be used, but this is straightforward
    var x1 >= 1, <= 5; 
    var x2 >= 1, <= 5; 
    var x3 >= 1, <= 5; 
    var x4 >= 1, <= 5;

    # specify the objective function
    minimize obj:
        x1 * x4 * (x1 + x2 + x3) + x3;
            
    # specify the constraints
    s.t.
      inequality:
        x1 * x2 * x3 * x4 >= 25;
                    
      equality:
        x1^2 + x2^2 + x3^2 +x4^2 = 40;

    # specify the starting point            
    let x1 := 1;
    let x2 := 5;
    let x3 := 5;
    let x4 := 1;

    # solve the problem
    solve;

    # print the solution
    display x1;
    display x2;
    display x3;
    display x4;

The line, `option solver ipopt;` tells AMPL to use \Ipopt
as the solver. The \Ipopt executable (installed in
\ref COMPILEINSTALL) must be in the `PATH` for AMPL to find it.
The remaining lines specify the problem in AMPL format. The problem can
now be solved by starting AMPL and loading the mod file:

    $ ampl
    > model hs071_ampl.mod;
    .
    .
    .

The problem will be solved using \Ipopt and the solution
will be displayed.

At this point, AMPL users may wish to skip the sections about
interfacing with code, but should read \ref OPTIONS and \ref OUTPUT.

\subsection INTERFACE_AMPL_CL Using Ipopt from the command line

It is possible to solve AMPL problems with \Ipopt directly
from the command line. However, this requires a file in format `.nl`
produced by ampl. If you have a model and data loaded in Ampl, you can
create the corresponding `.nl` file with name, say, `myprob.nl` by using the
Ampl command:

    write gmyprob

There is a small `.nl` file available in the \Ipopt
distribution. It is located at `Ipopt/test/mytoy.nl`. We use this file in
the remainder of this section. We assume that the file `mytoy.nl` is in
the current directory and that the command `ipopt` is a shortcut for
running the \Ipopt binary available in the `bin` directory of the
installation of \Ipopt.

We list below commands to perform basic tasks from the Linux prompt.

-   To solve mytoy.nl from the Linux prompt, use:

        ipopt mytoy

-   To see all command line options for \Ipopt, use:

        ipopt -=

-   To see more detailed information on all options for \Ipopt:

        ipopt mytoy 'print_options_documentation yes'

-   To run ipopt, setting the maximum number of iterations to 2 and
    print level to 4:

        ipopt mytoy 'max_iter 2 print_level 4'

If many options are to be set, they can be collected in a file `ipopt.opt`
that is automatically read by \Ipopt, if present, in the
current directory, see also \ref OPTIONS.

\section INTERFACE_CODE Interfacing with Ipopt through code

In order to solve a problem, \Ipopt needs more information
than just the problem definition (for example, the derivative
information). If you are using a modeling language like AMPL, the extra
information is provided by the modeling tool and the \Ipopt
interface. When interfacing with \Ipopt through your own
code, however, you must provide this additional information. The
following information is required by \Ipopt:

1.  Problem dimensions

    -   number of variables

    -   number of constraints

2.  Problem bounds

    -   variable bounds

    -   constraint bounds

3.  Initial starting point

    -   Initial values for the primal \f$x\f$ variables

    -   Initial values for the multipliers (only required for a warm start option)

4.  Problem Structure

    -   number of nonzeros in the Jacobian of the constraints

    -   number of nonzeros in the Hessian of the Lagrangian function

    -   sparsity structure of the Jacobian of the constraints

    -   sparsity structure of the Hessian of the Lagrangian function

5.  Evaluation of Problem Functions <BR/>
    Information evaluated using a given point (\f$x\f$, \f$\lambda\f$, \f$\sigma_f\f$ coming from \Ipopt)

    -   Objective function, \f$f(x)\f$

    -   Gradient of the objective, \f$\nabla f(x)\f$

    -   Constraint function values, \f$g(x)\f$

    -   Jacobian of the constraints, \f$\nabla g(x)^T\f$

    -   Hessian of the Lagrangian function,
        \f$\sigma_f \nabla^2 f(x) + \sum_{i=1}^m\lambda_i\nabla^2 g_i(x)\f$ <BR>
        (this is not required if a quasi-Newton options is chosen to
        approximate the second derivatives)

The problem dimensions and bounds are straightforward and come solely
from the problem definition. The initial starting point is used by the
algorithm when it begins iterating to solve the problem. If
\Ipopt has difficulty converging, or if it converges to a
locally infeasible point, adjusting the starting point may help.
Depending on the starting point, \Ipopt may also converge
to different local solutions.

Providing the sparsity structure of derivative matrices is a bit more
involved. \Ipopt is a nonlinear programming solver that is
designed for solving large-scale, sparse problems. While
\Ipopt can be customized for a variety of matrix formats,
the triplet format is used for the standard interfaces in this tutorial.
For an overview of the triplet format for sparse matrices, see
\ref TRIPLET. Before solving the problem,
\Ipopt needs to know the number of nonzero elements and the
sparsity structure (row and column indices of each of the nonzero
entries) of the constraint Jacobian and the Lagrangian function Hessian.
Once defined, this nonzero structure MUST remain constant for the entire
optimization procedure. This means that the structure needs to include
entries for any element that could ever be nonzero, not only those that
are nonzero at the starting point.

As \Ipopt iterates, it will need the values for problem functions
evaluated at particular points. Before we
can begin coding the interface, however, we need to work out the details
of these equations symbolically for example problem (HS071).

The gradient of the objective \f$f(x)\f$ is given by
\f[\left[
\begin{array}{c}
x_1 x_4 + x_4 (x_1 + x_2 + x_3) \\
x_1 x_4 \\
x_1 x_4 + 1 \\
x_1 (x_1 + x_2 + x_3)
\end{array}
\right]\f]
and the Jacobian of the constraints \f$g(x)\f$ is
\f[\left[
\begin{array}{cccc}
x_2 x_3 x_4     & x_1 x_3 x_4   & x_1 x_2 x_4   & x_1 x_2 x_3   \\
2 x_1           & 2 x_2         & 2 x_3         & 2 x_4
\end{array}
\right].
\f]

We also need to determine the Hessian of the Lagrangian.
(If a quasi-Newton option is chosen to approximate the second
derivatives, this is not required. However, if second derivatives
can be computed, it is often worthwhile to let \Ipopt
use them, since the algorithm is then usually more robust and
converges faster. More on the quasi-Newton approximation in
\ref QUASI_NEWTON.)
The Lagrangian function for the NLP (HS071) is defined as
\f$f(x) + g(x)^T \lambda\f$ and the Hessian of the Lagrangian function is,
technically, \f$\nabla^2 f(x_k) + \sum_{i=1}^m\lambda_i\nabla^2 g_i(x_k)\f$.
However, we introduce a factor (\f$\sigma_f\f$) in front of the objective
term so that \Ipopt can ask for the Hessian of the objective or the
constraints independently, if required. Thus, for \Ipopt
the symbolic form of the Hessian of the Lagrangian is
\f[ \sigma_f \nabla^2 f(x_k) + \sum_{i=1}^m\lambda_i\nabla^2 g_i(x_k) \f]
and for the example problem this becomes
\f[ \sigma_f \left[
\begin{array}{cccc}
2 x_4           & x_4           & x_4           & 2 x_1 + x_2 + x_3     \\
x_4             & 0             & 0             & x_1                   \\
x_4             & 0             & 0             & x_1                   \\
2 x_1+x_2+x_3   & x_1           & x_1           & 0
\end{array}
\right]
+
\lambda_1
\left[
\begin{array}{cccc}
0               & x_3 x_4       & x_2 x_4       & x_2 x_3       \\
x_3 x_4         & 0             & x_1 x_4       & x_1 x_3       \\
x_2 x_4         & x_1 x_4       & 0             & x_1 x_2       \\
x_2 x_3         & x_1 x_3       & x_1 x_2       & 0 
\end{array}
\right]
+
\lambda_2
\left[
\begin{array}{cccc}
2       & 0     & 0     & 0     \\
0       & 2     & 0     & 0     \\
0       & 0     & 2     & 0     \\
0       & 0     & 0     & 2
\end{array}
\right]\f] where the first term comes from the Hessian of the objective
function, and the second and third term from the Hessian of the
constraints. Therefore, the dual variables
\f$\lambda_1\f$ and \f$\lambda_2\f$ are the multipliers for the constraints.

The remaining sections of the tutorial will lead you through the coding
required to solve example problem (HS071) using, first C++, then C, and finally
Fortran. Completed versions of these examples can be found in
`$IPOPTDIR/Ipopt/examples` under `hs071_cpp`, `hs071_c`, and `hs071_f`.

As a user, you are responsible for coding two sections of the program
that solves a problem using \Ipopt: the main executable
(e.g., main) and the problem representation. Typically, you will write
an executable that prepares the problem, and then passes control over to
\Ipopt through an Optimize or Solve call. In this call, you
will give \Ipopt everything that it requires to call back
to your code whenever it needs functions evaluated (like the objective
function, the Jacobian of the constraints, etc.). In each of the three
sections that follow (C++, C, and Fortran), we will first discuss how to
code the problem representation, and then how to code the executable.

\subsection INTERFACE_CPP The C++ Interface

This tutorial assumes that you are familiar with the C++ programming
language, however, we will lead you through each step of the
implementation. For the problem representation, we will create a class
that inherits off of the pure virtual base class, Ipopt::TNLP.
For the executable (the main function) we will make the call to
\Ipopt through the Ipopt::IpoptApplication class.
In addition, we will also be using the
Ipopt::SmartPtr class which implements a reference counting
pointer that takes care of memory management (object deletion) for you
(for details, see \ref SMARTPTR).

After `make install` (see \ref COMPILEINSTALL), the header files that
define these classes are installed in `$PREFIX/include/coin-or`.

\subsubsection INTERFACE_CPP_NLP Coding the Problem Representation

We provide the required information by coding the HS071_NLP class, a
specific implementation of the TNLP base class. In the executable, we
will create an instance of the HS071_NLP class and give this class to
\Ipopt so it can evaluate the problem functions through the
Ipopt::TNLP interface. If you have any difficulty as the implementation
proceeds, have a look at the completed example in the
`Ipopt/examples/hs071_cpp` directory.

Start by creating a new directory `MyExample` under examples and create
the files hs071_nlp.hpp and hs071_nlp.cpp. In hs071_nlp.hpp, include
IpTNLP.hpp (the base class), tell the compiler that we are using the namespace
Ipopt, and create the declaration of the
HS071_NLP class, inheriting off of TNLP. Have a look at the Ipopt::TNLP class;
you will see eight pure virtual methods that we must
implement. Declare these methods in the header file. Implement each of
the methods in HS071_NLP.cpp using the descriptions given below. In
hs071_nlp.cpp, first include the header file for your class and tell
the compiler that you are using the namespace Ipopt. A
full version of these files can be found in the
`Ipopt/examples/hs071_cpp` directory.

It is very easy to make mistakes in the implementation of the function
evaluation methods, in particular regarding the derivatives.
\Ipopt has a feature that can help you to debug the
derivative code, using finite differences, see \ref DERIVCHECK.

Note that the return value of any bool-valued function should be true,
unless an error occurred, for example, because the value of a problem
function could not be evaluated at the required point.

\ref Ipopt::TNLP::get_nlp_info

<blockquote>
\snippet IpTNLP.hpp TNLP_get_nlp_info

\copydoc Ipopt::TNLP::get_nlp_info
</blockquote>

Our example problem has 4 variables (n), and 2 constraints (m). The
constraint Jacobian for this small problem is actually dense and has 8
nonzeros (we still need to represent this Jacobian using the sparse
matrix triplet format). The Hessian of the Lagrangian has 10 "symmetric"
nonzeros (i.e., nonzeros in the lower left triangular part.). Keep in
mind that the number of nonzeros is the total number of elements that
may *ever* be nonzero, not just those that are nonzero at the starting
point. This information is set once for the entire problem.

\snippet hs071_nlp.cpp TNLP_get_nlp_info


\ref Ipopt::TNLP::get_bounds_info

<blockquote>
\snippet IpTNLP.hpp TNLP_get_bounds_info

\copydoc Ipopt::TNLP::get_bounds_info
</blockquote>

In our example, the first constraint has a lower bound of 25 and no
upper bound, so we set the lower bound of constraint to 25 and the
upper bound to some number greater than 10<sup>19</sup>.
The second constraint is an equality constraint and we set both bounds to 40.
\Ipopt recognizes this as an equality constraint and does
not treat it as two inequalities.

\snippet hs071_nlp.cpp TNLP_get_bounds_info


\ref Ipopt::TNLP::get_starting_point

<blockquote>
\snippet IpTNLP.hpp TNLP_get_starting_point

\copydoc Ipopt::TNLP::get_starting_point
</blockquote>

In our example, we provide initial values for \f$x\f$ as specified in the
example problem. We do not provide any initial values for the dual
variables, but use an assert to immediately let us know if we are ever
asked for them.

\snippet hs071_nlp.cpp TNLP_get_starting_point


\ref Ipopt::TNLP::eval_f

<blockquote>
\snippet IpTNLP.hpp TNLP_eval_f

\copydoc Ipopt::TNLP::eval_f
</blockquote>

For our example, we ignore the `new_x` flag and calculate the objective.

\snippet hs071_nlp.cpp TNLP_eval_f


\ref Ipopt::TNLP::eval_grad_f

<blockquote>
\snippet IpTNLP.hpp TNLP_eval_grad_f

\copydoc Ipopt::TNLP::eval_grad_f
</blockquote>

In our example, we ignore the new\_x flag and calculate the values for
the gradient of the objective.

\snippet hs071_nlp.cpp TNLP_eval_grad_f


\ref Ipopt::TNLP::eval_g

<blockquote>
\snippet IpTNLP.hpp TNLP_eval_g

\copydoc Ipopt::TNLP::eval_g
</blockquote>

In our example, we ignore the new_x flag and calculate the values of
constraint functions.

\snippet hs071_nlp.cpp TNLP_eval_g


\ref Ipopt::TNLP::eval_jac_g

<blockquote>
\snippet IpTNLP.hpp TNLP_eval_jac_g

\copydoc Ipopt::TNLP::eval_jac_g
</blockquote>

In our example, the Jacobian is actually dense, but we still specify it
using the sparse format.

\snippet hs071_nlp.cpp TNLP_eval_jac_g


\ref Ipopt::TNLP::eval_h

<blockquote>
\snippet IpTNLP.hpp TNLP_eval_h

\copydoc Ipopt::TNLP::eval_h
</blockquote>

In our example, the Hessian is dense, but we still specify it using the
sparse matrix format. Because the Hessian is symmetric, we only need to
specify the lower left corner.

\snippet hs071_nlp.cpp TNLP_eval_h


\ref Ipopt::TNLP::finalize_solution

<blockquote>
This is the only method that is not mentioned in the beginning of this section.

\snippet IpTNLP.hpp TNLP_finalize_solution

\copydoc Ipopt::TNLP::finalize_solution
</blockquote>

In our example, we will print the values of some of the variables to the
screen.

\snippet hs071_nlp.cpp TNLP_finalize_solution

This is all that is required for our HS071_NLP class and the coding of
the problem representation.

\subsubsection INTERFACE_CPP_MAIN Coding the Executable

Now that we have a problem representation, the HS071_NLP class, we need
to code the main function that will call \Ipopt and ask
\Ipopt to find a solution.

Here, we must create an instance of our problem (HS071_NLP), create an
instance of the \Ipopt solver (Ipopt::IpoptApplication),
initialize it, and ask the solver to find a solution. We always use the
Ipopt::SmartPtr template class instead of raw C++ pointers when creating and
passing \Ipopt objects. To find out more information about
smart pointers and the SmartPtr implementation used in
\Ipopt, see \ref SMARTPTR.

Create the file `MyExample.cpp` in the `MyExample` directory. Include the
header files `HS071_NLP.hpp` and `IpIpoptApplication.hpp`, tell the
compiler to use the Ipopt namespace, and implement the main function.

\snippet hs071_main.cpp MAIN

The first line of code in `main()` creates an instance of HS071_NLP. We
then create an instance of the \Ipopt solver, Ipopt::IpoptApplication.
You could use `new` to create a new application object, but if you want
to make sure that your code would also work with a Windows DLL, you need
to use the factory, as done in the example above.
The call to `app->Initialize(...)` will initialize that object
and process the options (particularly the output related options). The
call to `app->OptimizeTNLP(...)` will run \Ipopt and try to
solve the problem. By default, \Ipopt will write its
progress to the console, and return the SolverReturn status.

\subsubsection INTERFACE_CPP_COMPILE Compiling and Testing the Example

Our next task is to compile and test the code. If you are familiar with
the compiler and linker used on your system, you can build the code,
telling the linker about the necessary libraries, as obtained via
`pkg-config --lflags ipopt`. The build system already created a sample
makefile. Copy `Ipopt/examples/hs071_cpp/Makefile` into your `MyExample`
directory. This makefile was created for the `hs071_cpp` code, but it can
be easily modified for your example problem. Edit the file, making the
following changes:

-   change the EXE variable \verbatim
EXE = my_example \endverbatim

-   change the OBJS variable \verbatim
OBJS = HS071_NLP.o MyExample.o \endverbatim

The code should compile easily with
\verbatim
$ make
\endverbatim

Now run the executable with
\verbatim
$ ./my_example
\endverbatim
and you should see output resembling the following:
\verbatim
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit https://github.com/coin-or/Ipopt
******************************************************************************

Number of nonzeros in equality constraint Jacobian...:        4
Number of nonzeros in inequality constraint Jacobian.:        4
Number of nonzeros in Lagrangian Hessian.............:       10

Total number of variables............................:        4
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        4
                     variables with only upper bounds:        0
Total number of equality constraints.................:        1
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        1
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.6109693e+01 1.12e+01 5.28e-01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.7410406e+01 8.38e-01 2.25e+01  -0.3 7.97e-01    -  3.19e-01 1.00e+00f  1
   2  1.8001613e+01 1.06e-02 4.96e+00  -0.3 5.60e-02   2.0 9.97e-01 1.00e+00h  1
   3  1.7199482e+01 9.04e-02 4.24e-01  -1.0 9.91e-01    -  9.98e-01 1.00e+00f  1
   4  1.6940955e+01 2.09e-01 4.58e-02  -1.4 2.88e-01    -  9.66e-01 1.00e+00h  1
   5  1.7003411e+01 2.29e-02 8.42e-03  -2.9 7.03e-02    -  9.68e-01 1.00e+00h  1
   6  1.7013974e+01 2.59e-04 8.65e-05  -4.5 6.22e-03    -  1.00e+00 1.00e+00h  1
   7  1.7014017e+01 2.26e-07 5.71e-08  -8.0 1.43e-04    -  1.00e-00 1.00e+00h  1
   8  1.7014017e+01 4.62e-14 9.09e-14  -8.0 6.95e-08    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 8

Number of objective function evaluations             = 9
Number of objective gradient evaluations             = 9
Number of equality constraint evaluations            = 9
Number of inequality constraint evaluations          = 9
Number of equality constraint Jacobian evaluations   = 9
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations             = 8
Total CPU secs in IPOPT (w/o function evaluations)   =      0.220
Total CPU secs in NLP function evaluations           =      0.000

EXIT: Optimal Solution Found.


Solution of the primal variables, x
x[0] = 1.000000e+00
x[1] = 4.743000e+00
x[2] = 3.821150e+00
x[3] = 1.379408e+00


Solution of the bound multipliers, z_L and z_U
z_L[0] = 1.087871e+00
z_L[1] = 2.428776e-09
z_L[2] = 3.222413e-09
z_L[3] = 2.396076e-08
z_U[0] = 2.272727e-09
z_U[1] = 3.537314e-08
z_U[2] = 7.711676e-09
z_U[3] = 2.510890e-09


Objective value
f(x*) = 1.701402e+01


*** The problem solved!
\endverbatim

This completes the basic C++ tutorial, but see \ref OUTPUT for
explanation on the standard console output of \Ipopt and
\ref OPTIONS for information about the use of options to customize the
behavior of \Ipopt.

The `Ipopt/examples/ScalableProblems` directory contains other NLP
problems coded in C++.

\subsubsection INTERFACE_CPP_ADDITIONAL Additional methods in TNLP

The following methods are available for additional features that are not
explained in the example. Default implementations for those methods are
provided, so that a user can safely ignore them, unless she wants to
make use of those features. From these features, only the intermediate
callback is already available in the C and Fortran interfaces.

\ref Ipopt::TNLP::intermediate_callback

<blockquote>
\snippet IpTNLP.hpp TNLP_intermediate_callback

\copydoc Ipopt::TNLP::intermediate_callback
</blockquote>


\ref Ipopt::TNLP::get_scaling_parameters

<blockquote>
\snippet IpTNLP.hpp TNLP_get_scaling_parameters

\copydoc Ipopt::TNLP::get_scaling_parameters
</blockquote>


\ref Ipopt::TNLP::get_number_of_nonlinear_variables

<blockquote>
This method is called only if the \ref QUASI_NEWTON "quasi-Newton approximation" is selected.

\snippet IpTNLP.hpp TNLP_get_number_of_nonlinear_variables

\copydoc Ipopt::TNLP::get_number_of_nonlinear_variables
</blockquote>


\ref Ipopt::TNLP::get_list_of_nonlinear_variables

<blockquote>
This method is called only if the \ref QUASI_NEWTON "quasi-Newton approximation" is selected.

\snippet IpTNLP.hpp TNLP_get_list_of_nonlinear_variables

\copydoc Ipopt::TNLP::get_list_of_nonlinear_variables
</blockquote>


\ref Ipopt::TNLP::get_variables_linearity

<blockquote>
\snippet IpTNLP.hpp TNLP_get_variables_linearity

\copydoc Ipopt::TNLP::get_variables_linearity
</blockquote>


\ref Ipopt::TNLP::get_constraints_linearity

<blockquote>
\snippet IpTNLP.hpp TNLP_get_constraints_linearity

\copydoc Ipopt::TNLP::get_constraints_linearity
</blockquote>


\ref Ipopt::TNLP::get_var_con_metadata

<blockquote>
\snippet IpTNLP.hpp TNLP_get_var_con_metadata

\copydoc Ipopt::TNLP::get_var_con_metadata
</blockquote>


\ref Ipopt::TNLP::finalize_metadata

<blockquote>
\snippet IpTNLP.hpp TNLP_finalize_metadata

\copydoc Ipopt::TNLP::finalize_metadata
</blockquote>


\ref Ipopt::TNLP::get_warm_start_iterate

<blockquote>
\snippet IpTNLP.hpp TNLP_get_warm_start_iterate

\copydoc Ipopt::TNLP::get_warm_start_iterate
</blockquote>


\subsection INTERFACE_C The C Interface

The C interface for \Ipopt is declared in the header file
IpStdCInterface.h, which is found in `$PREFIX/include/coin-or`;
while reading this section,
it will be helpful to have a look at this file.

In order to solve an optimization problem with the C interface, one has
to create an \ref IpoptProblem with the function \ref CreateIpoptProblem,
which later has to be passed to the \ref IpoptSolve function.
\ref IpoptProblem is a pointer to a C structure; you should not access
this structure directly, but only through the functions provided in the
C interface.

The \ref IpoptProblem created by \ref CreateIpoptProblem contains the problem
dimensions, the variable and constraint bounds, and the function
pointers for callbacks that will be used to evaluate the NLP problem
functions and their derivatives (see also the discussion of the C++
methods Ipopt::TNLP::get_nlp_info and Ipopt::TNLP::get_bounds_info for information
about the arguments of \ref CreateIpoptProblem).

The prototypes for the callback functions, \ref Eval_F_CB,
\ref Eval_Grad_F_CB, etc., are defined in the header file
IpStdCInterface.h. Their arguments correspond one-to-one to the
arguments for the corresponding C++ methods; for example,
for the meaning of `n`, `x`, `new_x`, `obj_value` in the declaration of
\ref Eval_F_CB see the discussion of Ipopt::TNLP::eval_f. The callback
functions should return `TRUE`, unless there was a problem doing the requested
function/derivative evaluation at the given point `x` (then it should
return `FALSE`).

Note the additional argument of type \ref UserDataPtr in the callback
functions. This pointer argument is available for you to communicate
information between the main program that calls \ref IpoptSolve and any of
the callback functions. This pointer is simply passed unmodified by
\Ipopt among those functions. For example, you can use this
to pass constants that define the optimization problem and are computed
before the optimization in the main C program to the callback functions.

After an \ref IpoptProblem has been created, you can set algorithmic options
for \Ipopt (see \ref OPTIONS) using the functions
\ref AddIpoptStrOption, \ref AddIpoptNumOption, and \ref AddIpoptIntOption.
Finally, the \Ipopt algorithm is called with \ref IpoptSolve, giving \Ipopt
the \ref IpoptProblem, the starting point, and arrays to store the solution
values (primal and dual variables), if desired. Finally, after
everything is done, \ref FreeIpoptProblem should be called to release
internal memory that is still allocated inside \Ipopt.

In the remainder of this section we discuss how the example problem (HS071)
can be solved using the C interface. A completed version of this example
can be found in `Ipopt/examples/hs071_c`.

In order to implement the example problem on your own, create a new
directory `MyCExample` and create a new file, `hs071_c.c`. Here, include
the interface header file IpStdCInterface.h, along with other necessary
header files, such as `stdlib.h` and `assert.h`. Add the prototypes and
implementations for the five callback functions. Have a look at the C++
implementation for `eval_f`, `eval_g`, `eval_grad_f`, `eval_jac_g`, and
`eval_h` in \ref INTERFACE_CPP_NLP. The C implementations have somewhat
different prototypes, but are implemented almost identically to the C++
code. See the completed example in `Ipopt/examples/hs071_c/hs071_c.c` if
you are not sure how to do this.

We now need to implement the main function, create the \ref IpoptProblem,
set options, and call \ref IpoptSolve. The \ref CreateIpoptProblem function
requires the problem dimensions, the variable and constraint bounds, and the
function pointers to the callback routines. The \ref IpoptSolve function
requires the \ref IpoptProblem, the starting point, and allocated arrays
for the solution. The main function from the example is shown next and
discussed below.

\snippet hs071_c.c MAIN

Here, we declare all the necessary variables and set the dimensions of
the problem. The problem has 4 variables, so we set n and allocate space
for the variable bounds (don't forget to call free for each of your
malloc calls before the end of the program). We then set the values for
the variable bounds.

The problem has 2 constraints, so we set m and allocate space for the
constraint bounds. The first constraint has a lower bound of 25 and no
upper bound. Here we set the upper bound to `2e19`. \Ipopt interprets any
number greater than or equal to the value of option \ref OPT_nlp_upper_bound_inf "nlp_upper_bound_inf"
as infinity. The default value of \ref OPT_nlp_lower_bound_inf "nlp_lower_bound_inf"
and \ref OPT_nlp_upper_bound_inf "nlp_upper_bound_inf" is `-1e19` and `1e19`,
respectively, and can be changed through \Ipopt options.
The second constraint is an equality with right hand side 40,
so we set both the upper and the lower bound to 40.

Next, we set the number of elements in the Jacobian and Hessian.
The Jacobian has 8 nonzero entries. For the Hessian we only specify the
number of nonzeros in the lower-triangular part, which is 10.
See \ref TRIPLET for a description of the sparse
matrix format.
Finally, we set the style for the row and column indices of the matrices
to 0 to indicate C-Style, that is, start indexing at 0.

We next create an instance of the \ref IpoptProblem by calling
\ref CreateIpoptProblem, giving it the problem dimensions and the variable
and constraint bounds. We also include the references to each of our
callback functions. \Ipopt uses these function pointers to
ask for evaluation of the NLP when required.

After freeing the bound arrays that are no longer required, the next three
lines illustrate how to change the value of options through the
interface. \Ipopt options can also be changed by creating an
`ipopt.opt` file (see \ref OPTIONS).
We next allocate space for the initial point and set the values as
given in the problem definition.

The call to \ref IpoptSolve can provide information about the
solution, but most of this is optional. Here, we want values for the
constraint and variable bound multipliers at the solution and thus
allocate space for these.

Further, we initialize the `user_data` that will be accessed in the
function evaluation callbacks.

We can now make the call to \ref IpoptSolve and find the solution of the
problem. We pass in the \ref IpoptProblem, the starting point `x`
(\Ipopt will use this array to return the solution or final
point as well). The next 5 arguments are pointers so \Ipopt
can fill in values at the solution. If these pointers are set to `NULL`,
\Ipopt will ignore that entry. For example, here we do not want the
constraint function values at the solution, so we set this entry to `NULL`.
We do want the value of the objective, and the multipliers
for the constraints and variable bounds. The last argument is a pointer
to our example-specific "user data". Any pointer that is passed in here
will be passed on to the callback functions.

The return code is an \ref ApplicationReturnStatus enumeration, see the
header file ReturnCodes_inc.h which is installed along
IpStdCInterface.h in the \Ipopt include directory.
After the optimizer terminates, we check the status and print the
solution if successful.

If the solve succeeded, we resolve a slightly modified version of the
problem. We use both the primal solution point and the dual multipliers
to warm-start \Ipopt for this second solve. Additionally, to improve
the warmstart, we reduce the amount by which \Ipopt pushes the starting
point into the interior of the variable bounds by setting the options
`bound_push` and `bound_frac`. If the solve was successful, we print the
solution again.

Finally, we free the \ref IpoptProblem and the
remaining memory and return from `main`.

\subsection INTERFACE_FORTRAN The Fortran Interface

The Fortran interface is essentially a wrapper of the \ref INTERFACE_C "C Interface".
The way to hook up \Ipopt in a Fortran program is very similar to how it
is done for the C interface, and the functions of the Fortran interface
correspond one-to-one to the those of the C and C++ interface, including
their arguments. You can find an implementation of the example problem
(HS071) in `$IPOPTDIR/Ipopt/examples/hs071_f`.

The only special things to consider are:

-   The return value of the function `IPCREATE` is of type `INTEGER` that
    must be large enough to capture a pointer on the particular machine.
    This means, that you have to declare the "handle" for the
    `IpoptProblem` as `INTEGER*8` if your program is compiled in 64-bit
    mode. All other `INTEGER`-type variables must be of the regular type.

-   For the call of `IPSOLVE` (which is the function that is to be called
    to run \Ipopt), all arrays, including those for the
    dual variables, must be given (in contrast to the C interface). The
    return value `IERR` of this function indicates the outcome of the
    optimization (see the include file `IpReturnCodes.inc` in the
    \Ipopt include directory).

-   The return value `IERR` of the remaining functions has to be set to
    zero, unless there was a problem during execution of the function
    call.

-   The callback functions (`EV_*` in the example) include the arguments
    `IDAT` and `DAT`, which are `INTEGER` and `DOUBLE PRECISION` arrays
    that are passed unmodified between the main program calling `IPSOLVE`
    and the evaluation subroutines `EV_*` (similarly to `UserDataPtr`
    arguments in the C interface). These arrays can be used to pass
    "private" data between the main program and the user-provided Fortran
    subroutines.

    The last argument of the `EV_*` subroutines, `IERR`, is to be set to 0
    by the user on return, unless there was a problem during the
    evaluation of the optimization problem function/derivative for the
    given point `X` (then it should return a non-zero value).

\subsection INTERFACE_JAVA The Java Interface JIpopt

This section is based on documentation by Rafael de Pelegrini Soares
(VRTech Industrial Technologies).

The Java Interface was written and contributed by Rafael de Pelegrini Soares
and later updated by Tong Kewei (Beihang University).
It offers an abstract base class \ref org.coinor.Ipopt "Ipopt"
with basic methods to specify an NLP, set a number of \Ipopt options,
to request \Ipopt to solve the NLP, and to retrieve a found
solution, if any. A HTML documentation of all available interface
methods of the \ref org.coinor.Ipopt "Ipopt" class can also be generated
via javadoc by executing `make javadoc` in the \Ipopt build directory
(`$IPOPTDIR`).

If the \Ipopt build includes the Java interface, then a JAR file
`org.coinor.ipopt.jar` containing class org.coinor.Ipopt will have been
installed in `$PREFIX/lib`. To use the Java interface, make sure that this
jar file is part of your Java classpath.
A sample makefile that compiles and runs example HS071 can be found in
`$IPOPTDIR/examples/hs071_java`.

In the following, we discuss necessary steps to implement example (HS071)
with `JIpopt`.
We create a new Java source file HS071.java and define a class
`HS071` that extends the class \ref org.coinor.Ipopt "Ipopt".
In the class constructor, we call the \ref org.coinor.Ipopt::create()
method of `JIpopt`, which works analogously to \ref Ipopt::TNLP::get_nlp_info()
of the C++ interface. It initializes an \ref Ipopt::IpoptApplication object
and informs `JIpopt` about the problem size (number of variables, constraints,
nonzeros in Jacobian and Hessian).

\snippet HS071.java HS071

Next, we add callback functions that are called by `JIpopt`
to obtain variable bounds, constraint sides, and a starting point:

\snippet HS071.java get_bounds_info

\snippet HS071.java get_starting_point

In the following, we implement the evaluation methods in a way that is
very similar to the C++ interface:

\snippet HS071.java eval

Finally, we add a main routine to run this example. The main routines
creates an instance of our class, calls the solve method
\ref org.coinor.Ipopt::OptimizeNLP "OptimizeNLP", and prints the solution.

\snippet HS071.java main

The \ref org.coinor.Ipopt::OptimizeNLP "OptimizeNLP" method returns the
\Ipopt solve status as integer, which indicates whether the problem was
solved successfully. Further, the methods
\ref org.coinor.Ipopt::getObjectiveValue() "getObjectiveValue()",
\ref org.coinor.Ipopt::getVariableValues() "getVariableValues()",
\ref org.coinor.Ipopt::getConstraintMultipliers() "getConstraintMultipliers()",
\ref org.coinor.Ipopt::getLowerBoundMultipliers() "getLowerBoundMultipliers()", and
\ref org.coinor.Ipopt::getUpperBoundMultipliers() "getUpperBoundMultipliers()"
can be used to obtain the objective value, the primal solution value of
the variables, and dual solution values.

\subsection INTERFACE_R The R Interface ipoptr

This section is based on documentation by Jelmer Ypma
(University College London).

The `ipoptr` package (see also \ref INSTALL_R) offers a R function
`ipoptr` which takes an NLP specification, a starting point, and \Ipopt
options as input and returns information about an \Ipopt run
(status, message, ...) and a solution point.

In the following, we discuss necessary steps to implement example (HS071)
with `ipoptr`. A more detailed documentation of `ipoptr` is
available in [contrib/RInterface/inst/doc/ipoptr.pdf](https://github.com/coin-or/Ipopt/raw/master/contrib/RInterface/inst/doc/ipoptr.pdf).

First, we define the objective function and its gradient
\code{.unparsed}
> eval_f <- function( x ) { 
    return( x[1]*x[4]*(x[1] + x[2] + x[3]) + x[3] ) 
  }
> eval_grad_f <- function( x ) {
    return( c( x[1] * x[4] + x[4] * (x[1] + x[2] + x[3]),
               x[1] * x[4],
               x[1] * x[4] + 1.0,
               x[1] * (x[1] + x[2] + x[3]) ) )       
  }
\endcode

Then we define a function that returns the value of the two constraints.
We define the bounds of the constraints (in this case the `g_L` and
`g_U` are 25 and 40) later.
\code{.unparsed}
> # constraint functions
> eval_g <- function( x ) {
     return( c( x[1] * x[2] * x[3] * x[4],
                x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 ) )
  }
\endcode

Then we define the structure of the Jacobian, which is a dense matrix in
this case, and function to evaluate it
\code{.unparsed}
> eval_jac_g_structure <- list( c(1,2,3,4), c(1,2,3,4) )
> eval_jac_g <- function( x ) {
     return( c ( x[2]*x[3]*x[4],
                 x[1]*x[3]*x[4],
                 x[1]*x[2]*x[4],
                 x[1]*x[2]*x[3],
                 2.0*x[1],
                 2.0*x[2],
                 2.0*x[3],
                 2.0*x[4] ) )
  }
\endcode

The Hessian is also dense, but it looks slightly more complicated
because we have to take into account the Hessian of the objective
function and of the constraints at the same time, although you could
write a function to calculate them both separately and then return the
combined result in `eval_h`.
\code{.unparsed}
> # The Hessian for this problem is actually dense, 
> # This is a symmetric matrix, fill the lower left triangle only.
> eval_h_structure <- list( c(1), c(1,2), c(1,2,3), c(1,2,3,4) )
> eval_h <- function( x, obj_factor, hessian_lambda ) {
     values <- numeric(10)
     values[1] = obj_factor * (2*x[4]) # 1,1

     values[2] = obj_factor * (x[4])   # 2,1
     values[3] = 0                     # 2,2
     
     values[4] = obj_factor * (x[4])   # 3,1
     values[5] = 0                     # 4,2
     values[6] = 0                     # 3,3
     
     values[7] = obj_factor * (2*x[1] + x[2] + x[3]) # 4,1
     values[8] = obj_factor * (x[1])                 # 4,2
     values[9] = obj_factor * (x[1])                 # 4,3
     values[10] = 0                                  # 4,4
     
     # add the portion for the first constraint
     values[2] = values[2] + hessian_lambda[1] * (x[3] * x[4]) # 2,1
         
     values[4] = values[4] + hessian_lambda[1] * (x[2] * x[4]) # 3,1
     values[5] = values[5] + hessian_lambda[1] * (x[1] * x[4]) # 3,2
     
     values[7] = values[7] + hessian_lambda[1] * (x[2] * x[3]) # 4,1
     values[8] = values[8] + hessian_lambda[1] * (x[1] * x[3]) # 4,2
     values[9] = values[9] + hessian_lambda[1] * (x[1] * x[2]) # 4,3
    
     # add the portion for the second constraint
     values[1] = values[1] + hessian_lambda[2] * 2 # 1,1
     values[3] = values[3] + hessian_lambda[2] * 2 # 2,2
     values[6] = values[6] + hessian_lambda[2] * 2 # 3,3
     values[10] = values[10] + hessian_lambda[2] * 2 # 4,4
     
     return ( values )
  }
\endcode

After the hard part is done, we only have to define the initial values,
the lower and upper bounds of the control variables, and the lower and
upper bounds of the constraints. If a variable or a constraint does not
have lower or upper bounds, the values `-Inf` or `Inf` can be used. If
the upper and lower bounds of a constraint are equal, \Ipopt recognizes
this as an equality constraint and acts accordingly.
\code{.unparsed}
> # initial values
> x0 <- c( 1, 5, 5, 1 )
> # lower and upper bounds of control
> lb <- c( 1, 1, 1, 1 )
> ub <- c( 5, 5, 5, 5 )
> # lower and upper bounds of constraints
> constraint_lb <- c(  25, 40 )
> constraint_ub <- c( Inf, 40 )
\endcode

Finally, we can call \Ipopt with the `ipoptr` function. In
order to redirect the \Ipopt output into a file, we use
\Ipopt's and options.
\code{.unparsed}
> opts <- list("print_level" = 0,
               "file_print_level" = 12,
               "output_file" = "hs071_nlp.out")
> print( ipoptr( x0 = x0, 
                 eval_f = eval_f, 
                 eval_grad_f = eval_grad_f, 
                 lb = lb, 
                 ub = ub, 
                 eval_g = eval_g, 
                 eval_jac_g = eval_jac_g, 
                 constraint_lb = constraint_lb, 
                 constraint_ub = constraint_ub, 
                 eval_jac_g_structure = eval_jac_g_structure,
                 eval_h = eval_h,
                 eval_h_structure = eval_h_structure,
                 opts = opts) )

Call:

ipoptr(x0 = x0, eval_f = eval_f, eval_grad_f = eval_grad_f, lb = lb, 
  ub = ub, eval_g = eval_g, eval_jac_g = eval_jac_g,
  eval_jac_g_structure = eval_jac_g_structure, constraint_lb = constraint_lb,
  constraint_ub = constraint_ub, eval_h = eval_h, eval_h_structure = eval_h_structure,
  opts = opts)

Ipopt solver status: 0 ( SUCCESS: Algorithm terminated 
successfully at a locally optimal point, satisfying the 
convergence tolerances (can be specified by options). )

Number of Iterations....: 8 
Optimal value of objective function:  17.0140171451792 
Optimal value of controls: 1 4.743 3.82115 1.379408
\endcode

To pass additional data to the evaluation routines, one can either
supply additional arguments to the user defined functions and `ipoptr` or
define an environment that holds the data and pass this environment to
`ipoptr`. Both methods are shown in the file `tests/parameters.R` that
comes with `ipoptr`.

As a very simple example, suppose we want to find the minimum of
\f$f(x) = a_1 x^2 + a_2 x + a_3\f$ for different values of the
parameters \f$a_1\f$, \f$a_2\f$ and \f$a_3\f$.

First, we define the objective function and its gradient using, assuming
that there is some variable `params` that contains the values of the
parameters.
\code{.unparsed}
> eval_f_ex1 <- function(x, params) { 
     return( params[1]*x^2 + params[2]*x + params[3] ) 
  }
> eval_grad_f_ex1 <- function(x, params) { 
     return( 2*params[1]*x + params[2] ) 
  }
\endcode
Note, that the first parameter should always be the control variable.
All of the user-defined functions should contain the same set of
additional parameters. You have to supply them as input argument to all
functions, even if you're not using them in some of the functions.

Then we can solve the problem for a specific set of parameters, in this
case \f$a_1=1\f$, \f$a_2=2\f$, and \f$a_3=3\f$, from initial value
\f$x_0=0\f$, with the following command
\code{.unparsed}
> # solve using ipoptr with additional parameters
> ipoptr(x0          = 0, 
         eval_f      = eval_f_ex1, 
         eval_grad_f = eval_grad_f_ex1,
         opts        = list("print_level"=0),
         params      = c(1,2,3) )
             
Call:

ipoptr(x0 = 0, eval_f = eval_f_ex1, eval_grad_f = eval_grad_f_ex1, 
    opts = list(print_level = 0), params = c(1, 2, 3))


Ipopt solver status: 0 ( SUCCESS: Algorithm terminated 
successfully at a locally optimal point, satisfying the 
convergence tolerances (can be specified by options). )

Number of Iterations....: 1 
Optimal value of objective function:  2 
Optimal value of controls: -1
\endcode

For the second method, we don't have to supply the parameters as
additional arguments to the function.
\code{.unparsed}
> eval_f_ex2 <- function(x) { 
     return( params[1]*x^2 + params[2]*x + params[3] ) 
  }
> eval_grad_f_ex2 <- function(x) { 
     return( 2*params[1]*x + params[2] ) 
  }
\endcode

Instead, we define an environment that contains specific values of `params`:
\code{.unparsed}
> # define a new environment that contains params
> auxdata        <- new.env()
> auxdata$params <- c(1,2,3)
\endcode

To solve this we supply `auxdata` as an argument to `ipoptr`, which will
take care of evaluating the functions in the correct environment, so
that the auxiliary data is available.
\code{.unparsed}
> # pass the environment that should be used to evaluate functions to ipoptr
> ipoptr(x0                 = 0, 
         eval_f             = eval_f_ex2, 
         eval_grad_f        = eval_grad_f_ex2, 
         ipoptr_environment = auxdata,
         opts               = list("print_level"=0) )
             
Call:

ipoptr(x0 = 0, eval_f = eval_f_ex2, eval_grad_f = eval_grad_f_ex2, 
    opts = list(print_level = 0), ipoptr_environment = auxdata)


Ipopt solver status: 0 ( SUCCESS: Algorithm terminated 
successfully at a locally optimal point, satisfying the 
convergence tolerances (can be specified by options). )

Number of Iterations....: 1 
Optimal value of objective function:  2 
Optimal value of controls: -1
\endcode

*/
