It should be noted that optimization of a convex bounded lipschitz function is exactly what most modern statistical learning (AI) models are based on.
Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.
I'd push back on this. Most of the core optimization techniques (eg, ADAM, stochastic gradient descent) are straight out of the convex optimization literature. Generally you need to use optimizers that work well on convex objectives because near minimizers, functions tend to be convex. (Proof by contradiction: a non-convex point has a strict descent direction.)
The fact that neural networks are highly nonconvex has encouraged a lot of research, but it's more of the kind aimed at resolving tension: these methods are probably good for convex functions, why do they continue to work for nonconvex problems, and are there tweaks we can make to improve them in that setting? It's not a lot of de novo theory; more standing on the shoulders of giants, etc etc.
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What do you mean by this? A neural network hypothesis space is not typically strictly convex or a lipschitz function.