How can I use Optim/BFGS on functions I can't evaluate but whose gradients I can compute?

Specific Domains Optimization (Mathematical)
11

Here’s what I checked, minimizing |x|^2 with a fixed step-size of 0.1. There’s no error, but if I just have my target function return e.g. 0, you can see it terminates early even with a negative f_tol,

using Optim, LineSearches, LinearAlgebra

bfgs = BFGS(
    alphaguess = LineSearches.InitialStatic(alpha=0.1), 
    linesearch = LineSearches.Static()
)

# do a proper minimization
optimize(
    x->dot(x,x),
    x->2x,
    ones(10),
    bfgs,
    inplace=false
).iterations # returns 181

# attempt to always return 0 from the target function,
# but with correct gradient
# this doesn't work and will terminate early
optimize(
    x->0,
    x->2x,
    ones(10),
    bfgs,
    Optim.Options(f_tol=-1, allow_f_increases=true),
    inplace=false
).iterations # returns 2

# the only workaround I've found 
f₀ = 0
optimize(
    x->(global f₀-=1),
    x->2x,
    ones(10),
    bfgs,
    inplace=false
).iterations # returns 181