add `ADNLSModel` constructor in `arglina`

[tmigot]

This is an example of how we could use ADNLSModel for the least square objective.

Currently, the tests break because in the JuMP models currently implemented we generally don't have the 1/2 factor in front of the objective. #162

@abelsiqueira @dpo Any opinion on this?

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[tmigot]

@dpo Here is an example of the benefit of having both in-place and out-place residual with the same name, so that we can change the backend effortlessly.

using ADNLPModels, NLPModels, ReverseDiff
function arglina(; n::Int = default_nvar, type::Val{T} = Val(Float64), kwargs...) where {T}
  function F(r, x)
    m = 2 * n
    for i=1:n
      r[i] = x[i] - T(2 / m) * sum(x[j] for j = 1:n) - 1
      r[i + n] = -T(2 / m) * sum(x[j] for j = 1:n) - 1
    end
    return r
  end
  function F(x)
    r = similar(x, 2 * n)
    return F(r, x)
  end
  x0 = ones(T, n)
  return ADNLPModels.ADNLSModel(F, x0, 2 * n, name = "arglina"; kwargs...)
end

nlp = arglina(n = 10)
F = nlp.F
output = typeof(nlp.meta.x0)(undef, nlp.nls_meta.nequ)
input = nlp.meta.x0

# Use ForwardDiff on x -> nlp.F(x)
jac_residual(nlp, input)
@show @allocated jac_residual(nlp, input) # 6528

# Use ReverseDiff on (r, x) -> nlp.F(r, x)
cfJ = ReverseDiff.JacobianTape(nlp.F, output, input)
ReverseDiff.jacobian!(cfJ, input)
@show @allocated ReverseDiff.jacobian!(cfJ, input) # 1808 !!

# Use ReverseDiff on (r, x) -> nlp.F(r, x) and pre-allocate the result
result = zeros(20, 10)
ReverseDiff.jacobian!(result, cfJ, input)
@show @allocated ReverseDiff.jacobian!(result, cfJ, input) # 0

[dpo]

Hi @tmigot. This is quite old and I forget what you were trying to explain. I don't see anything in the example above that would be complicated if the in-place function were called F!. What am I missing?

[tmigot]

The issue is that the function returns

> ADNLPModels.ADNLSModel(F, x0, 2 * n, name = "arglina"; kwargs...)

so that wouldn't work if we have an F and an F!.

[tmigot]

I wouldn't merge it anyway because right now calling obj for a NLS allocates, while we were trying to make some efforts along this direction in here https://github.com/JuliaSmoothOptimizers/OptimizationProblems.jl/pull/241