Hello everybody,

**A little background:**

I’m working on a package that implements creates differentiable McCormick relaxations of functions. These relaxations are generally tighter than interval bounds and since they are convex/concave in construction they are useful in a branch-and-bound optimization framework. The base object is for these are SMC:

```
struct SMC{T} <: AbstractMC{T}
cc::T # concave relaxation value
cv::T # convex relaxation value
Intv::Interval{T} # interval over which the relaxation is defined
end
```

A library of operations is then constructed in a typical operator overloading fashion. So if one wanted to compute the relaxations of exp(-x^2)*sin(x) at x = 2 in the interval [1,3], you’d use the following code:

```
x = SMC{Float64}(2,2,Interval(1,3))
y = exp(-x^2)*sin(x)
convex = y.cv
concave = y.cc
```

I’d like to be able to get gradients of y.cc and y.cv via automatic differentiation (forward or mixed type). So ultimately, I’d like to setup functions like below and use a differentiation tool to calculate gradients.

```
using ForwardDiff
using IntervalArithmetic
function example(x::Vector)
Xbox = IntervalBox(-1..1,-2..2)
temp1 = SMC{Float64}(x[1],x[1],Xbox[1])
temp2 = SMC{Float64}(x[1],x[1],Xbox[1])
temp3 = exp(temp1*temp2)
temp3.cv
end
g = x -> ForwardDiff.gradient(example, x);
x0 = [1.0,2.0]
g(x0) $ gradient
```

**Is this type of automatic differentiation feasible with ForwardDiff in concept?**

I’ve been trying to get this style working for a bit and I’ve run into a lot of conversion errors with regard to Dual Numbers. So I just want to make sure this is possible before I wade deeper into this problem.

I’m almost ready to publish this package and I’ve got a separate package which includes gradient objects in SMC and naively does forward differentiation. So I can share that shortly.

Thanks for the time and consideration.