I am trying to find the minimizer \theta of the following nonlinear and non-differentiable function: g(H(X | \theta),X). Here, X is known and imported from dataset; \theta=(\theta_1,\theta_2,\theta_3) are the parameters I want to solve for; H is discrete and therefore makes g(\cdot) non-differentiable with respect to \theta. There is only one simple constraint such that the first parameter \theta_1 is positive.

Since the argument H is discrete, I wonder if I can still use automatic differentiation in Ipopt for this optimization problem? Thank you.

Well, yes and no. Automatic differentiation can handle discrete functions, but all it will do is (correctly) tell you that the derivative is zero:

julia> using ForwardDiff
[ Info: Precompiling ForwardDiff [f6369f11-7733-5829-9624-2563aa707210]
funct
julia> function f(x)
if x > 0
return 1
else
return 2
end
end
f (generic function with 1 method)
julia> ForwardDiff.derivative(f, 1.0)
0

So using autodiff with an optimization method that expects a gradient is unlikely to work well. Perhaps one of the non-gradient-based algorithms from NLOpt would be more appropriate here? https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/

I think my function is different from this. The function g(\cdot) is kind of smooth, although argument H is discrete. In your example, function f(\cdot) is discrete but argument x is smooth.

But the function you are dealing with is f(\theta) = g(H(X \mid \theta), X) which in general will not be continuous(ly differentiable) if H isn’t, even if g is.

Depending on how bad things are, stochastic (eg simulated annealing) or robust deterministic derivative-free optimization methods are used in these cases. See

Thanks for your comments. I have read these two posts previously. My problem is also simulated method of moments. I was testing blackbox and showed these two posts to my friend. But he still wants me to test Ipopt + user-defined numerical derivatives in Julia.

BTW, do you happen to know this new global optimizer named TikTak, which is developed by Fatih Guvenen?

I do not think it will apply to my case. Basically, it only supports univariate function. I am using MathOptInterface instead and call Ipopt with self-defined derivatives.