That is indeed how the subject has been developed historically and often introduced in optimisation courses. First one studies Linear Programs (minimize a linea

That is indeed how the subject has been developed historically and often introduced in optimisation courses. First one studies Linear Programs (minimize a linear function with affine equality and affine inequality constraints), then Quadratic Programs (quadratic functions with affine constraints), Quadratically Constrained Quadratic Programs, etc. So we often assume very specific properties of not just the objective but also the constraints, and we get some extra juice in these cases. Through this process of seeing what results we can prove in more generality, we finally arrived at getting a good body of theory and algorithms for minimizing convex functions with convex constraints. In some courses they start with the general theory, as the top down approach can save some time and brain space.