### Run cyipopt example

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Commands to run a cyipopt example after successful installation.

```bash
$ cd test
$ python -c "import cyipopt"
$ python examplehs071.py
```

--------------------------------

### Build and Link Commands

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Example commands showing the build and linking process for cyipopt.c, including compiler flags and library paths.

```bash
$ python setup.py build
...
    x86_64-linux-gnu-gcc -pthread -DNDEBUG -g -fwrapv -O2 -Wall -Wstrict-prototypes -fno-strict-aliasing
      -Wdate-time -D_FORTIFY_SOURCE=2 -g -fdebug-prefix-map=/build/python2.7-3hk45v/python2.7-2.7.15~rc1=. 
      -fstack-protector-strong -Wformat -Werror=format-security -fPIC
      -I/usr/local/include/coin  # points to Ipopt headers
      -I/usr/local/include/coin/ThirdParty  # points to Ipopt third party headers
      -I/usr/local/include/python2.7/numpy/core/include  # points to NumPy headers
      -I/usr/include/python2.7  # points to Python 2.7 headers
      -c src/cyipopt.c -o build/temp.linux-x86_64-2.7/src/cyipopt.o
    x86_64-linux-gnu-gcc -pthread -shared -Wl,-O1 -Wl,-Bsymbolic-functions -Wl,-Bsymbolic-functions -Wl,-z,relro
      -fno-strict-aliasing -DNDEBUG -g -fwrapv -O2 -Wall -Wstrict-prototypes -Wdate-time -D_FORTIFY_SOURCE=2 -g
      -fdebug-prefix-map=/build/python2.7-3hk45v/python2.7-2.7.15~rc1=. -fstack-protector-strong -Wformat
      -Werror=format-security -Wl,-Bsymbolic-functions -Wl,-z,relro -Wdate-time -D_FORTIFY_SOURCE=2 -g
      -fdebug-prefix-map=/build/python2.7-3hk45v/python2.7-2.7.15~rc1=. -fstack-protector-strong -Wformat
      -Werror=format-security build/temp.linux-x86_64-2.7/src/cyipopt.o
      -L/usr/local/lib
      -L/lib/../lib
      -L/usr/lib/../lib
      -L/usr/lib/gcc/x86_64-linux-gnu/5
      -L/usr/lib/gcc/x86_64-linux-gnu/5/../../..
      -L/usr/lib/gcc/x86_64-linux-gnu/5/../../../../lib
      -L/usr/lib/gcc/x86_64-linux-gnu/5/../../../x86_64-linux-gnu
      -lipopt -llapack -lblas -lm -ldl -lcoinmumps -lblas -lgfortran -lm -lquadmath  # linking to relevant libs
      -lcoinhsl -llapack -lblas -lgfortran -lm -lquadmath -lcoinmetis  # linking to relevant libs
      -o build/lib.linux-x86_64-2.7/cyipopt.so
    ...

```

--------------------------------

### pip install cyipopt[docs,examples,tests]

Source: https://github.com/mechmotum/cyipopt/blob/master/CHANGELOG.rst

Example of installing cyipopt with pip extras for documentation, examples, and tests.

```bash
pip install cyipopt[docs,examples,tests]
```

--------------------------------

### Build and Install HSL

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Build and install HSL after configuration.

```bash
(hsl-test) $ make
   (hsl-test) $ make install
```

--------------------------------

### Install dependencies on Ubuntu 22.04 using APT

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs all required dependencies for cyipopt using Ubuntu's APT package manager.

```sh
$ apt install build-essential pkg-config python3-pip python3-dev cython3 python3-numpy coinor-libipopt1v5 coinor-libipopt-dev
```

--------------------------------

### Build Ipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Commands to build Ipopt from source.

```bash
$ mkdir $IPOPTDIR/build
$ cd $IPOPTDIR/build
$ ../configure
$ make
$ make test
```

--------------------------------

### Compile cyipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to build cyipopt from source.

```bash
$ python setup.py build
```

--------------------------------

### Install cyipopt from source on Linux and macOS

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Builds and installs cyipopt from source after dependencies are met.

```sh
$ python -m pip install .
```

--------------------------------

### Install build dependencies on Windows using Pip

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs necessary build tools and libraries for cyipopt on Windows using Pip.

```powershell
> python -m pip install numpy cython setuptools
```

--------------------------------

### Install dependencies on Ubuntu 18.04 using APT

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs dependencies for cyipopt on Ubuntu 18.04 using APT.

```sh
$ sudo apt install build-essential pkg-config python-dev cython python-numpy coinor-libipopt1v5 coinor-libipopt-dev
```

--------------------------------

### Install cyipopt from a local copy on Ubuntu 22.04

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs cyipopt from a local source directory after APT dependencies are installed.

```sh
$ cd /cyipopt/source/directory/
$ python3 -m pip install .
```

--------------------------------

### Install cyipopt from PyPi on Ubuntu 22.04

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs the cyipopt package from PyPi after APT dependencies are installed.

```sh
$ python3 -m pip install cyipopt
```

--------------------------------

### Install cyipopt from source on Windows using Conda's Ipopt binary

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs Ipopt from Conda Forge and then builds cyipopt from source.

```powershell
> conda.exe install -c conda-forge ipopt
> python -m pip install --no-deps --no-build-isolation .
```

--------------------------------

### Install cyipopt Package

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to install the cyipopt package into the Python environment.

```bash
$ python setup.py install
```

--------------------------------

### Install System Dependencies on Ubuntu

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Commands to install necessary system dependencies on Ubuntu 18.04 for compiling Ipopt.

```bash
$ sudo apt install pkg-config python-dev wget
$ sudo apt build-dep coinor-libipopt1v5

$ sudo apt install python-pip

$ pip install --user numpy cython
```

--------------------------------

### Configure cyipopt with HSL

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Set HSL linear solver options in cyipopt.

```python
problem.add_option('linear_solver', 'ma57')
   problem.add_option('hsllib', 'libcoinhsl.so')
```

--------------------------------

### Install cyipopt on Windows

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Create and activate a conda environment for cyipopt.

```bash
$ conda create -n hsl-test -c conda-forge cyipopt
   $ conda activate hsl-test
```

--------------------------------

### Download and extract cyipopt source

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Commands to download and extract the cyipopt source code.

```bash
$ cd ~
$ wget https://files.pythonhosted.org/packages/05/57/a7c5a86a8f899c5c109f30b8cdb278b64c43bd2ea04172cbfed721a98fac/ipopt-0.1.9.tar.gz
$ tar -xvf ipopt-0.1.8.tar.gz
$ cd ipopt
```

--------------------------------

### Solving the Constrained Optimization Problem

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Example of setting up constraints and solving the problem using minimize_ipopt.

```python
from scipy.optimize import rosen, rosen_der, rosen_hess

constr = {'type': 'ineq', 'fun': con, 'jac': con_jac, 'hess': con_hess}

# initial guess
x0 = np.array([1.1, 1.1, 1.1, 1.1, 1.1])

# solve the problem
res = minimize_ipopt(rosen, jac=rosen_der, hess=rosen_hess, x0=x0, constraints=constr)
```

--------------------------------

### Build cyipopt from source on Ubuntu 18.04

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Compiles cyipopt from source on Ubuntu 18.04 after installing dependencies.

```sh
python setup.py build
```

--------------------------------

### Install cyipopt using conda

Source: https://github.com/mechmotum/cyipopt/blob/master/README.rst

Recommended installation method for cyipopt on Linux, Mac, and Windows using conda.

```bash
conda install -c conda-forge cyipopt
```

--------------------------------

### HS071 Problem Setup and Solve

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Sets up the bounds, constraints, initial guess, and instantiates the HS071 problem, then solves it using CyIpopt.

```python
lb = [1.0, 1.0, 1.0, 1.0]
ub = [5.0, 5.0, 5.0, 5.0]

cl = [25.0, 40.0]
cu = [2.0e19, 40.0]

x0 = [1.0, 5.0, 5.0, 1.0]

nlp = HS071(
    n=len(x0),
    m=len(cl),
    lb=lb,
    ub=ub,
    cl=cl,
    cu=cu,
)

x, info = nlp.solve(x0)

```

--------------------------------

### SciPy Compatible Interface Example

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Demonstrates the usage of the minimize_ipopt function, which mimics scipy.optimize.minimize.

```python
from scipy.optimize import rosen, rosen_der
from cyipopt import minimize_ipopt
x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
res = minimize_ipopt(rosen, x0, jac=rosen_der)
print(res)
```

--------------------------------

### Check cyipopt linking with ldd

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to verify that cyipopt.so is correctly linked.

```bash
$ ldd build/lib.linux-x86_64-2.7/cyipopt.so
```

--------------------------------

### Install cyipopt using Conda

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs cyipopt and its dependencies from the Conda Forge channel.

```sh
$ conda install -c conda-forge cyipopt
```

--------------------------------

### Install development version

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/development.rst

Installs a development version of cyipopt in editable mode.

```sh
(cyipopt-dev)$ python -m pip install --no-deps --no-build-isolation -e .
```

--------------------------------

### Download Ipopt Source Code

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to download the Ipopt source code using wget.

```bash
$ cd ~
$ wget https://www.coin-or.org/download/source/Ipopt/Ipopt-3.12.11.tgz
```

--------------------------------

### Configure HSL

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Configure HSL to install into the current environment, link against environment's BLAS and LAPACK, and use environment's compilers.

```bash
(hsl-test) $ cd /path/to/coinhsl-2014.01.10/
   (hsl-test) $ ./configure \
      --prefix=/home/<username>/miniconda/envs/hsl-test/ \
      --with-blas="-L/home/<username>/miniconda/envs/hsl-test/lib/ -lblas" \
      LIBS="-llapack" \
      FC=/home/<username>/miniconda/envs/hsl-test/bin/gfortran \
      CC=/home/<username>/miniconda/envs/hsl-test/bin/gcc \

```

--------------------------------

### Install build dependencies on Windows using Conda

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Installs necessary build tools and libraries for cyipopt on Windows using Conda.

```powershell
> conda.exe install -c conda-forge numpy cython setuptools pkg-config
```

--------------------------------

### Create Conda environment with HSL and cyipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to create a Conda environment with gfortran and cyipopt.

```bash
$ conda create -n hsl-test -c conda-forge gfortran cyipopt
$ conda activate hsl-test
```

--------------------------------

### Configure cyipopt on Windows

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Add HSL solver option in cyipopt on Windows.

```python
problem.add_option('linear_solver', 'ma57')
```

--------------------------------

### Set LD_LIBRARY_PATH for cyipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Export command to set the LD_LIBRARY_PATH for Ipopt.

```bash
$ export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:~/Ipopt-3.12.11/build/lib
```

--------------------------------

### Set environment variables for Ipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Environment variables to set after building Ipopt.

```bash
$ export IPOPT_PATH="~/Ipopt-3.12.11/build"
$ export PKG_CONFIG_PATH=$PKG_CONFIG_PATH:$IPOPT_PATH/lib/pkgconfig
$ export PATH=$PATH:$IPOPT_PATH/bin
```

--------------------------------

### Set Ipopt Directory Environment Variable

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to set a temporary environment variable pointing to the Ipopt installation directory.

```bash
$ export IPOPTDIR=~/Ipopt-3.12.11
```

--------------------------------

### Add LD_LIBRARY_PATH to .bashrc

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to add the LD_LIBRARY_PATH export to the .bashrc file for persistence.

```bash
$ echo 'export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:$HOME/Ipopt-3.12.11/build/lib' >> ~/.bashrc
```

--------------------------------

### Adding an option

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Example of adding an option to the NLP problem.

```python
nlp.add_option('tol', 1e-7)
```

--------------------------------

### Build manylinux wheels for linux/amd64

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Build manylinux wheels for a tagged version of cyipopt for linux/amd64 platform using docker.

```bash
docker run -v $(pwd):/wheels --rm --platform=linux/amd64 quay.io/pypa/manylinux_2_28_x86_64 bash /wheels/build_manylinux_wheels.sh GIT_TAG
```

--------------------------------

### Build manylinux wheels for linux/aarch64

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Build manylinux wheels for a tagged version of cyipopt for linux/aarch64 platform using docker.

```bash
docker run -v $(pwd):/wheels --rm --platform=linux/aarch64 quay.io/pypa/manylinux_2_28_aarch64 bash /wheels/build_manylinux_wheels.sh GIT_TAG
```

--------------------------------

### Verify HSL Linking

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Check if the HSL shared library is properly linked, especially BLAS.

```bash
(hsl-test) $ ldd ~/miniconda/envs/hsl-test/lib/libcoinhsl.so
```

--------------------------------

### Check Shared Library Dependencies

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to check the shared library dependencies of the compiled cyipopt.so file using ldd.

```bash
$ ldd build/lib.linux-x86_64-2.7/cyipopt.so
linux-vdso.so.1 (0x00007ffc1677c000)
libipopt.so.0 => /usr/local/lib/libipopt.so.0 (0x00007fcdc8668000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007fcdc8277000)
libcoinmumps.so.0 => /usr/local/lib/libcoinmumps.so.0 (0x00007fcdc7eef000)
libcoinhsl.so.0 => /usr/local/lib/libcoinhsl.so.0 (0x00007fcdc7bb4000)
liblapack.so.3 => /usr/lib/x86_64-linux-gnu/liblapack.so.3 (0x00007fcdc732e000)
libblas.so.3 => /usr/lib/x86_64-linux-gnu/libblas.so.3 (0x00007fcdc70d3000)
libdl.so.2 => /lib/x86_64-linux-gnu/libdl.so.2 (0x00007fcdc6ecf000)
libstdc++.so.6 => /usr/lib/x86_64-linux-gnu/libstdc++.so.6 (0x00007fcdc6b46000)
libm.so.6 => /lib/x86_64-linux-gnu/libm.so.6 (0x00007fcdc67a8000)
/lib64/ld-linux-x86-64.so.2 (0x00007fcdc8d20000)
libgcc_s.so.1 => /lib/x86_64-linux-gnu/libgcc_s.so.1 (0x00007fcdc6590000)
libcoinmetis.so.0 => /usr/local/lib/libcoinmetis.so.0 (0x00007fcdc6340000)
libgfortran.so.3 => /usr/lib/x86_64-linux-gnu/libgfortran.so.3 (0x00007fcdc600f000)
libopenblas.so.0 => /usr/lib/x86_64-linux-gnu/libopenblas.so.0 (0x00007fcdc3d69000)
libgfortran.so.4 => /usr/lib/x86_64-linux-gnu/libgfortran.so.4 (0x00007fcdc398a000)
libquadmath.so.0 => /usr/lib/x86_64-linux-gnu/libquadmath.so.0 (0x00007fcdc374a000)
libpthread.so.0 => /lib/x86_64-linux-gnu/libpthread.so.0 (0x00007fcdc352b000)

```

--------------------------------

### Extract Ipopt Source Code

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to extract the downloaded Ipopt source code archive.

```bash
$ tar -xvf Ipopt-3.12.11.tgz
```

--------------------------------

### Check Ipopt linking in Conda environment

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/install.rst

Command to check what Ipopt is linked against within a Conda environment.

```bash
(hsl-test) $ ldd ~/miniconda/envs/hsl-test/lib/libipopt.so
```

--------------------------------

### Build documentation

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/development.rst

Commands to build the HTML documentation.

```sh
(cyipopt-dev)$ cd docs
(cyipopt-dev)$ make html
```

--------------------------------

### Importing Libraries

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Import necessary libraries from JAX and cyipopt, and configure JAX for 64-bit precision and CPU usage.

```python
from jax import config

# Enable 64 bit floating point precision
config.update("jax_enable_x64", True)

# We use the CPU instead of GPU und mute all warnings if no GPU/TPU is found.
config.update('jax_platform_name', 'cpu')

import jax.numpy as np
from jax import jit, grad, jacfwd, jacrev
from cyipopt import minimize_ipopt
```

--------------------------------

### Clone the repository

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/development.rst

Steps to clone the cyipopt repository.

```sh
$ git clone git@github.com:mechmotum/cyipopt.git
$ cd cyipopt
```

--------------------------------

### Instantiate the Problem class

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Create an instance of the cyipopt.Problem class with the defined problem parameters.

```python
nlp = cyipopt.Problem(
   n=len(x0),
   m=len(cl),
   problem_obj=HS071(),
   lb=lb,
   ub=ub,
   cl=cl,
   cu=cu,
)
```

--------------------------------

### Building Derivatives and JIT Compiling

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

JIT compile the objective and constraint functions, and then build their derivatives (gradient, Jacobian, Hessian) using JAX's automatic differentiation capabilities.

```python
# jit the functions
obj_jit = jit(objective)
con_eq_jit = jit(eq_constraints)
con_ineq_jit = jit(ineq_constrains)

# build the derivatives and jit them
obj_grad = jit(grad(obj_jit))  # objective gradient
obj_hess = jit(jacrev(jacfwd(obj_jit))) # objective hessian
con_eq_jac = jit(jacfwd(con_eq_jit))  # jacobian
con_ineq_jac = jit(jacfwd(con_ineq_jit))  # jacobian
con_eq_hess = jacrev(jacfwd(con_eq_jit)) # hessian
con_eq_hessvp = jit(lambda x, v: con_eq_hess(x) * v[0]) # hessian vector-product
con_ineq_hess = jacrev(jacfwd(con_ineq_jit))  # hessian
con_ineq_hessvp = jit(lambda x, v: con_ineq_hess(x) * v[0]) # hessian vector-product
```

--------------------------------

### Calling minimize_ipopt

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Set up the constraints, initial guess, and bounds, then call cyipopt's minimize_ipopt function to solve the optimization problem.

```python
# constraints
cons = [
    {'type': 'eq', 'fun': con_eq_jit, 'jac': con_eq_jac, 'hess': con_eq_hessvp},
    {'type': 'ineq', 'fun': con_ineq_jit, 'jac': con_ineq_jac, 'hess': con_ineq_hessvp}
 ]

# starting point
x0 = np.array([1.0, 5.0, 5.0, 1.0])

# variable bounds: 1 <= x[i] <= 5
bnds = [(1, 5) for _ in range(x0.size)]

# executing the solver
res = minimize_ipopt(obj_jit, jac=obj_grad, hess=obj_hess, x0=x0, bounds=bnds,
                     constraints=cons, options={'disp': 5})
```

--------------------------------

### Import cyipopt and numpy

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Before using cyipopt, you need to import it along with numpy.

```python
import cyipopt
import numpy as np
```

--------------------------------

### Executing the solver

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Calling the solve method to run the optimization algorithm.

```python
x, info = nlp.solve(x0)
```

--------------------------------

### Define bounds and initial guess

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Define the lower and upper bounds for variables and constraints, as well as an initial guess for the variables.

```python
lb = [1.0, 1.0, 1.0, 1.0]
ub = [5.0, 5.0, 5.0, 5.0]

cl = [25.0, 40.0]
cu = [2.0e19, 40.0]

x0 = [1.0, 5.0, 5.0, 1.0]
```

--------------------------------

### Add optimization options

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Set optimization parameters by calling the add_option method.

```python
nlp.add_option('mu_strategy', 'adaptive')
```

--------------------------------

### Create and activate Conda environment

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/development.rst

Commands to create a Conda environment using the provided yml file and activate it.

```sh
$ conda env create -f cyipopt-dev.yml
$ conda activate cyipopt-dev
```

--------------------------------

### Run tests

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/development.rst

Command to run the test suite using pytest.

```sh
(cyipopt-dev)$ pytest
```

--------------------------------

### Constraint and Sparse Jacobian Implementation

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Implementation of constraint function and its sparse Jacobian using scipy.sparse.coo_array.

```python
from scipy.sparse import coo_array

def con(x):
    return np.array([ 10 -x[1]**2 - x[2], 100.0 - x[4]**2 ])

def con_jac(x):
    # Dense Jacobian:
    # J = (0  -2*x[1]   -1   0     0     )
    #     (0   0         0   0   -2*x[4] )
    # Sparse Jacobian (COO)
    rows = np.array([0, 0, 1])
    cols = np.array(([1, 2, 4]))
    data = np.array([-2*x[1], -1, -2*x[4]])
    return coo_array((data, (rows, cols)))
```

--------------------------------

### Constraint Hessian Implementation

Source: https://github.com/mechmotum/cyipopt/blob/master/docs/source/tutorial.rst

Implementation of the Hessian-vector-product for constraints, considering the Lagrangian function.

```python
def con_hess(x, _lambda):
    H1 = np.array([
        [0,  0, 0, 0, 0],
        [0, -2, 0, 0, 0 ],
        [0,  0, 0, 0, 0 ],
        [0,  0, 0, 0, 0 ],
        [0,  0, 0, 0, 0 ]
    ])

    H2 = np.array([
        [0, 0, 0, 0,  0],
        [0, 0, 0, 0,  0],
        [0, 0, 0, 0,  0],
        [0, 0, 0, 0,  0],
        [0, 0, 0, 0, -2]
    ])
    return _lambda[0] * H1 + _lambda[1] * H2
```