Clarify the documentation on user-defined gradients (#2639)

docs/src/manual/nlp.md

@@ -204,7 +204,7 @@ julia> expr = @NLexpression(model, sin(x) + 1)
 
 With the exception of the splatting syntax discussed below, all expressions
 must be simple scalar operations. You cannot use `dot`, matrix-vector products,
-vector slices, etc. 
+vector slices, etc.
 ```jldoctest nlp_scalar_only; setup=:(model = Model(); @variable(model, x[1:2]); @variable(model, y); c = [1, 2])
 julia> @NLobjective(model, Min, c' * x + 3y)
 ERROR: Unexpected array [1 2] in nonlinear expression. Nonlinear expressions may contain only scalar expressions.
@@ -262,7 +262,7 @@ In addition to this list of functions, it is possible to register custom
     a function is not needed. Two exceptions are if you want to provide custom
     derivatives, or if the function is not available in the scope of the
     nonlinear expression.
-    
+
 !!! warning
     User-defined functions must return a scalar output. For a work-around, see
     [User-defined functions with vector outputs](@ref).
@@ -395,7 +395,7 @@ vector as the first argument that is filled in-place:
 ```@example
 using JuMP #hide
 f(x, y) = (x - 1)^2 + (y - 2)^2
-function ∇f(g::Vector{T}, x::T, y::T) where {T}
+function ∇f(g::AbstractVector{T}, x::T, y::T) where {T}
     g[1] = 2 * (x - 1)
     g[2] = 2 * (y - 2)
     return
@@ -407,7 +407,9 @@ register(model, :my_square, 2, f, ∇f)
 @NLobjective(model, Min, my_square(x[1], x[2]))
 ```
 
-Hessian information is not supported for multivariate functions.
+!!! warning
+    Make sure the first argument to `∇f` supports an `AbstractVector`, and do
+    not assume the input is `Float64`.
 
 ### Register a function, gradient, and hessian
 

src/nlp.jl

@@ -1861,16 +1861,16 @@ end
 Register the user-defined function `f` that takes `dimension` arguments in
 `model` as the symbol `s`. In addition, provide a gradient function `∇f`.
 
-The functions `f`and ∇f must support all subtypes of `Real` as arguments. Do not
-assume that the inputs are `Float64`.
+The functions `f`and `∇f` must support all subtypes of `Real` as arguments. Do
+not assume that the inputs are `Float64`.
 
 ## Notes
 
  * If the function `f` is univariate (i.e., `dimension == 1`), `∇f` must return
    a number which represents the first-order derivative of the function `f`.
  * If the function `f` is multi-variate, `∇f` must have a signature matching
-   `∇f(g::Vector{T}, args::T...) where {T<:Real}`, where the first argument is a
-   vector `g` that is modified in-place with the gradient.
+   `∇f(g::AbstractVector{T}, args::T...) where {T<:Real}`, where the first
+   argument is a vector `g` that is modified in-place with the gradient.
  * If `autodiff = true` and `dimension == 1`, use automatic differentiation to
    compute the second-order derivative information. If `autodiff = false`, only
    first-order derivative information will be used.
@@ -1892,7 +1892,7 @@ register(model, :foo, 1, f, ∇f; autodiff = true)
 model = Model()
 @variable(model, x[1:2])
 g(x::T, y::T) where {T<:Real} = x * y
-function ∇g(g::Vector{T}, x::T, y::T) where {T<:Real}
+function ∇g(g::AbstractVector{T}, x::T, y::T) where {T<:Real}
     g[1] = y
     g[2] = x
     return