Quite a bit of optimization is easy to reason about in linear algebra. Take linear and mixed integer programming, for example. And convex optimization subsumes linear optimization in general. There is a lot of nonlinear optimization, but I can assure you with extremely high confidence that the common thread you're seeing here isn't duality, but more abstractly linearity.
Likewise cyclic things show up all the time in purely algebraic (read: discrete, non-smooth) contexts. We have that in vector spaces, group theory, rings, modules, etc.
The canonical example is robotic motion and the reason why Lie theory is used there. You have very discrete states (positions) that you want to interpolate between smoothly.
Likewise cyclic things show up all the time in purely algebraic (read: discrete, non-smooth) contexts. We have that in vector spaces, group theory, rings, modules, etc.