Recently I've read some stuff about incremental optimization problems, but I can't see what's the difference between those and online optimization problems. My impression is that I can define every online problem as an incremental counterpart (the reverse is clearly true).
Here go the (not very formal) definitions. In an incremental problem, one is given a sequence of instances of an optimization problem. The (i+1)-th instance is an "extension" of the i-th one. The (i+1)-th solution must be calculated without knowledge of the "future" instances, and has to keep the decisions made at the i-th solution. The classical example is with the k-median problem: after opening k facilities, one wants to have k' > k facilities but does not want to demolish the old ones.
In an online problem, (the usual definition is that) one is given a sequence of "requests". Here, one also has to answer a request without knowledge of the future requests. One wants to optimize the cost/gain of answering the sequence as a whole.
I believe that for any online problem I can define an "offline" optimization problem that fits the incremental definition (and what I usually see is the reverse). If the definitions are equivalent, what is the point of using a different name for the same concept?