## Derivative Checker

When writing code for the evaluation of derivatives it is very easy to make mistakes (much easier than writing it correctly the first time :)). As a convenient feature, IPOPT provides the option to run a simple derivative checker, based on finite differences, before the optimization is started.

To use the derivative checker, you need to use the option derivative_test. By default, this option is set to none, i.e., no finite difference test is performed, It is set to first-order, then the first derivatives of the objective function and the constraints are verified, and for the setting second-order, the second derivatives are tested as well.

The verification is done by a simple finite differences approximation, where each component of the user-provided starting point is perturbed one of the other. The relative size of the perturbation is determined by the option derivative_test_perturbation. The default value (, about the square root of the machine precision) is probably fine in most cases, but if you believe that you see wrong warnings, you might want to play with this parameter. When the test is performed, IPOPT prints out a line for every partial derivative, for which the user-provided derivative value deviates too much from the finite difference approximation. The relative tolerance for deciding when a warning should be issued, is determined by the option derivative_test_tol. If you want to see the user-provided and estimated derivative values with the relative deviation for each single partial derivative, you can switch the derivative_test_print_all option to yes.

A typical output is:

```Starting derivative checker.

* grad_f[          2] = -6.5159999999999991e+02    ~ -6.5559997134793468e+02  [ 6.101e-03]
* jac_g [    4,    4] =  0.0000000000000000e+00    ~  2.2160643690464592e-02  [ 2.216e-02]
* jac_g [    4,    5] =  1.3798494268463347e+01 v  ~  1.3776333629422766e+01  [ 1.609e-03]
* jac_g [    6,    7] =  1.4776333636790881e+01 v  ~  1.3776333629422766e+01  [ 7.259e-02]

Derivative checker detected 4 error(s).
```

The star (```*`'') in the first column indicates that this line corresponds to some partial derivative for which the error tolerance was exceeded. Next, we see which partial derivative is concerned in this output line. For example, in the first line, it is the second component of the objective function gradient (or the third, if the C_STYLE numbering is used, i.e., when counting of indices starts with 0 instead of 1). The first floating point number is the value given by the user code, and the second number (after ```~`'') is the finite differences estimation. Finally, the number in square brackets is the relative difference between these two numbers.

For constraints, the first index after jac_g is the index of the constraint, and the second one corresponds to the variable index (again, the choice of the numbering style matters).

Since also the sparsity structure of the constraint Jacobian has to be provided by the user, it can be faulty as well. For this, the `` v'' after a user-provided derivative value indicates that this component of the Jacobian is part of the user provided sparsity structure. If there is no ``v'', it means that the user did not include this partial derivative in the list of non-zero elements. In the above output, the partial derivative ``jac_g[4,4]'' is non-zero (based on the finite difference approximation), but it is not included in the list of non-zero elements (missing ``v''), so that the user probably made a mistake in the sparsity structure. The other two Jacobian entries are provided in the non-zero structure but their values seem to be off.

For second derivatives, the output looks like:

```*             obj_hess[    1,    1] =  1.8810000000000000e+03 v  ~  1.8820000036612328e+03  [ 5.314e-04]
*     3-th constr_hess[    2,    4] =  1.0000000000000000e+00 v  ~  0.0000000000000000e+00  [ 1.000e+00]
```

There, the first line shows the deviation of the user-provided partial second derivative in the Hessian for the objective function, and the second line show an error in a partial derivative for the Hessian of the third constraint (again, the numbering style matters).

Since the second derivatives are approximates by finite differences of the first derivatives, you should first correct errors for the first derivatives. Also, since the finite difference approximations are quite expensive, you should try to debug a small instance of your problem if you can.

Another useful option is derivative_test_first_index which allows your to start the derivative test with variables with a larger index.Finally, it is of course always a good idea to run your code through some memory checker, such as valgrind on Linux.