What's the difference between online and incremental optimization?
This is discussed in section 2.2.3 of Jeffrey Hartline's thesis: http://www.cs.cornell.edu/w8/~jhartlin/finaldiss.pdf

Online problems are all about informational uncertainty: you don't know what inputs are coming tomorrow, and the difficulty is often information theoretic, not computational. On the other hand, there is no uncertainty in an incremental optimization problem as Hartline defines it: every parameter of the problem is known at the outset. Without computational restrictions, the problems can always be solved optimally.

So perhaps your definition is wrong, since indeed yours sounds like an online problem. The problem of "incremental optimization" seems to be defined in this 2008 thesis, and differs from your definition in that there is no uncertainty.