Robust or global multivariate rootfinder

[dpsanders]

Yes it can; see e.g.

[Ann] IntervalRootFinding.jl for finding all roots of a multivariate function

We are excited to announce v0.2 of the IntervalRootFinding.jl package, which has been completely rewritten from scratch. This package exports a function roots that is guaranteed to find all roots of a given function \mathbb{R}^n \to \mathbb{R}^n in a given box X \subseteq \mathbb{R}^n (or tell you that it is unable to do so). Functions are written with standard Julia syntax; no macros or other magic are required! It is able to do this thanks to the power of interval arithmetic methods, based …

The docs are about to be refreshed once we work out how to deploy them.

If your function is polynomial, you can try HomotopyContinuation.jl.
If it is Julia code and only uses “standard” functions like sin, and no control flow, you can try IntervalRootFinding.jl.

In your problem statement, I don’t understand what you mean by “robust”, nor the “global root”. If you actually have an optimisation problem, rather than a root finding problem, and with the same caveats as before, you can try IntervalOptimisation.jl (or indeed any of the optimization packages available through JuMP).