Add PyNumero Reduced Hessian example

[blnicho]

Documenting #1282 as an issue instead of a PR.

Motivation:

Implement a basic example that demonstrates how to compute the reduced hessian of a nonlinear equality constrained problem with pynumero. Extensions to general nonlinear optimization problems with bounds can be implemented in pynumero following a very similar approach where the KKT matrix is the augmented system of the barrier subproblem at the optimal point. @carldlaird @blnicho @michaelbynum please refer to A.35 in Here for the computation of the KKT matrix.

The calculation of the reduced hessian can be beneficial for different applications of pyomo models. These include advanced step NMPC (e.g. pyomo.dae @blnicho ) and estimation of covariance matrices in parameter estimation problems (e.g. Parmest @kaklise, KIPET @salvadorgarciamunoz). Note that now that pynumero is supported on windows the implementation of the reduced hessian can be done in pynumero without the need of compiling any additional packages

@carldlaird for PyNumero itself, efficient calculation of the reduced hessian can be very useful for designing new algorithms. An interesting example is presented in here. There information from the reduced hessian is used within the inertia correction.

[adowling2]

@michaelbynum @carldlaird @blnicho

For my class, I tried replicating Example 4.20 in Biegler (2010). Here is a notebook with the example:
https://ndcbe.github.io/CBE60499/04.05-Second-Order.html

This is a two-variable optimization problem with a single inequality constraint. The calculated reduced Hessian has dimensions 2x2. We expect a 1x1 reduced Hessian because the constraint removes a degree of freedom.

Changing the constraint from an inequality (which is strongly active) to an equality constraint does not change the KKT multiplier (as expected) but does change the inverse reduced Hessian!

With an inequality constraint, the calculated reduced Hessian is:

[[ 0.50075352 -0.15638396]
 [-0.15638396  0.36113557]]

With an equality constraint, the calculated reduced Hessian is:

[[ 0.21115307 -0.26122822]
 [-0.26122822  0.32317873]]

Summary of problems:

• Wrong dimensions for the (inverse) reduced Hessian.
• (Inverse) reduced Hessian changes when switching a strongly active constraint from inequality to equality.
• Calculate does not match textbook