No. Integer programming in the operations research sense is "just" solving linear optimization problems with the constraint that the solution variables need to

No. Integer programming in the operations research sense is "just" solving linear optimization problems with the constraint that the solution variables need to be integer. In particular, IP doesn't really involve the multiplicative structure of the integers, which is crucial for how Gödel proved his results. Also, I'd point out that while there's significant overlap between integer programming and discrete optimization, they're not the same thing. Integer programming is one tool used in discrete optimization among many.