Adds VectorNonlinearOracle support

[andrewrosemberg]

Adds VectorNonlinearOracle support by creating a bridge for the nonlinear backend

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.88%. Comparing base (ca1e88a) to head (cff5092).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #334      +/-   ##
==========================================
+ Coverage   91.19%   91.88%   +0.68%     
==========================================
  Files          16       17       +1     
  Lines        2509     2722     +213     
==========================================
+ Hits         2288     2501     +213     
  Misses        221      221              

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[andrewrosemberg]

There is only one issue. if I create a model with diff_optimizer it defaults to use conic backend even though it is not compatible with vno. Trying to figure out how to avoid this.

[joaquimg]

I'd like @blegat 's eyes on it.
For now, can we move the tests to seperate file only for VNO?

[blegat]

The way to transform duals is by making sure the term in the x gradient of the Lagrangian function stays the same.
Say you transform x in {f(x) = 0} with dual λ into f(x) in {0} with dual μ.
The terms in the Lagrangian are ⟨λ, x⟩ and ⟨μ, f(x)⟩.
Their derivative w.r.t. x are λ and ∇f(x)' * μ. So given the dual μ, we compute λ by multiplying by the transpose jacobian, this is what is done in jump-dev/MathOptInterface.jl#2890
Here, we should invert the transposed Jacobian.