[measurablefunc]

You should look into condition numbers & how that applies to numerical stability of discretized optimization. If you take a continuous formulation & naively discretize you might get lucky & get a convergent & stable implementation but more often than not you will end up w/ subtle bugs & instabilities for ill-conditioned initial conditions.

[phreeza]

I understand that much, but it seems like "your naive timestep may need to be smaller than you think or you need to do some extra work" rather than the more fundamental objection from OP?

[measurablefunc]

The translation from continuous to discrete is not automatic. There is a missing verification in the linked analysis. The mapping must be verified for stability for the proper class of initial/boundary conditions. Increasing the resolution from 64 bit floats to 128 bit floats doesn't automatically give you a stable discretized optimizer from a continuous formulation.

[phyalow]

Or you can just try stuff and see if it works

[measurablefunc]

Point still stands, translation from continuous to discrete is not as simple as people think.

[phreeza]

Numerical issues totally exist but the reason has nothing to do with the fact that Cauchy sequences don't exist on a computer imo.

[measurablefunc]

The abstract formulation is different from the concrete implementation. It is precisely b/c the abstractions do not exist on computers that the abstract analysis does not automatically transfer the necessary analytical properties to the digital implementation. Cauchy sequences & Dedekind cuts are abstract & do not exist on digital computers.