This is a non-convex global optimisation problem. The state-of-the-art way to solve this is to use a factorable relaxation.
A key insight here is that $e^{-\alpha X}$ is convex (since your $\alpha$ is positive).
The methodology would be as follows:
- Introduce a new auxiliary variable $w=e^{-\alpha X}$
- You now have $Z=Yw$, and $w=e^{-\alpha X}$
- Because both constraints are non-convex, you split each to two inequalities:
$Z\leq Yw, -Z\leq -Yw$
$ w\leq e^{-\alpha X}, -w\leq -e^{-\alpha X}$
The first set of inequalities can be convexified using a McCormick relaxation.
The second set of inequalities are convex and concave respectively. The convex inequality can be relaxed using an outer approximation, and the concave inequality using a secant.
You then plug your relaxed problem into a branch-and-bound algorithm and it will converge quadratically.
Note that this methodology is independent of the signs of $Z,Y,X$.
Alternatively, you can plug this into a global solver which will do all of this for you automatically. Couenne is an open-source choice, and if you are an academic/student you can also use SCIP or our own Octeract Engine for free.