Basically, k-means is an algorithm. You give it data and the number of clusters you want to find in the data. It finds these clusters and returns their centers (known as centroids) and possibly assigns each data point to a cluster. "Optimizing" a k-means algorithm doesn't make much sense, IMO. What you probably want to say is that you ran the algorithm and got some centroids.

If you run k-means with new data but tell it to use particular centroids (that you got from a previous run of k-means), then it'll use these centroids as starting points and update them to match the new data.

1. Feed the algorithm some data.
2. It internally chooses initial centroids. How to choose these very first centroids isn't a simple problem. They're usually chosen "randomly". For example, you can pick K distinct points from your dataset.
3. K-means then does its thing and adjusts these initial centroids to best fit your data. This happens in several iterations.
4. Finally, these adjusted centroids are returned.
5. Now you put in new data and the centroids from the previous step.
   1. If the number of iterations is zero, there's nothing to be done, so the centroids remain unchanged.
   2. If the number of iterations is greater than zero, K-means performs these iterations and adjusts these centroids to better fit the new data.
6. The new, potentially adjusted centroids are returned.

Basically, k-means will adjust the centroids you give it in such a way that these centroids define clusters that describe your data well enough. When you run k-means with no centroids, it'll generate random centroids for you.