Hello fellow Julia users,

I want to ask a question using ForwardDiff.jl.

```
using LinearAlgebra, Statistics, Distributions, Random
using Optim, ForwardDiff
using Optim: converged, maximum, maximizer, minimizer, iterations
#%% DGP
Random.seed!(1)
N = 100000;
β0 = 0.15; β1 = 0.25; β2 = -1.5;
σ = 3.0;
x1 = randn(N,1);
x2 = randn(N,1);
ε = σ * randn(N,1);
ydata = β0 .+ β1.*x1 .+ β2.*x2 .+ ε;
X = [ones(N,1) x1 x2]; # Nx3 matrix
β_hat = inv(X'X)*X'ydata
#%%
function normaleqn(β,ydata,X)
N = size(X)[1];
f = zeros(eltype(β),size(X));
A = zeros(eltype(β),size(β));
for i=1:N
f[i,:] = X[i,:] * (ydata[i] .- X[i,:]' * β) # 1x3 * (1x1)
end
A = 1/N*sum(f,dims=1)'
end
#%%
β = ones(3,1)
ortho(β) = normaleqn(β,ydata,X)
jaco = ForwardDiff.jacobian(β->ortho(β),zeros(3,1)) # for testing
obj_I(β) = ortho(β)' * I * ortho(β)
obj_I(ones(3,1)) # for testing
opt = optimize(obj_I,zeros(3,1),BFGS(),autodiff=:forward,Optim.Options(iterations=1200));
```

The last line returns an error message,

```
**MethodError: no method matching extract_gradient!(::Type{ForwardDiff.Tag{typeof(obj_I),Float64}}, ::DiffResults.MutableDiffResult{1,Float64,Tuple{Array{Float64,2}}}, ::Array{ForwardDiff.Dual{ForwardDiff.Tag{typeof(obj_I),Float64},Float64,3},2})**
```

I now at least have a sense that ForwardDiff.jl uses the Dual Type and so I need to provide `eltype()`

to let Julia flexibly define the types for the returned object. So, I managed to get the jacobian of `ortho(β)`

, as shown in the middle of the script. However, I still can’t get the optimization routine run. Can someone explain why this is happening, please?

Thank you for your attention.