Efficient Array Normalization
Given that at least N "operations" are needed since each array element is affected, you can't get the number of operations down by much. It's not clear what your complexity measure exactly is – why is sum += array[i] a single operation? What about the loop? The actual CPU needs to increment i and compare it to array.size(), there are jumps involved and so on. The operation sum += array[i] is probably more than one instruction. Not to mention things you actually forgot to count, like sum = 0.

That said, I can't see how to improve on your algorithm, and I don't think it's possible, though as mentioned above, the notion of complexity is very shaky here, so perhaps there is a "trick" which will work under your complexity measure, whatever it is.