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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 ...

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Assuming the continuous relaxation is convex you can most likely use conic optimization with the exponential cone. The Mosek modelling cookbook has the details. Unsurprisingly Mosek can solve the mixed integer version of such problems.

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You may find a partial answer to your question in the following article (forthcoming in OR) by JP Vielma and J Huchette https://arxiv.org/abs/1708.00050 In that paper, the authors consider the problem of approximating non-linear functions of one or two variables in the objective via the disjunction of multiple hyperplanes. You can then pass the resulting MIP ...

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Welcome to OR Stack Exchange. Your question is not clear. You are interested in special points in discrete optimization spaces. But you describe a mixed-variable problem involving both continuous variables $x$ and integer variables $y$. For a given $y$, the theory of calculus of variations applies to the continuous subproblem on $x$. But I don't think you ...

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If a deterministic global optimisation solver (such as Baron) reports a local solution, that solution is reliable. If the solver is terminated prematurely, the global solver will return the best solution it has found so far. For NLP, it is quite common that global solvers find the global solution very early on, and then spend the majority of time proving it ...

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