Previous answers essentially present Jenks as a special case of K-means. However, this source makes an important distinction: K-means solely "searches for minimum distance between data points and the centers of clusters they belong to". Jenks takes this objective and adds a penalty for the proximity between the centers of clusters, and thus it also searches "for maximum difference between cluster centers themselves".
The logic is that, even if two clusters are internally very compact, they may be hard to distinguish when their centers are very close.
Thus, for $n$ data points and $k$ clusters, K-means would minimize $C$:
$$ C = \sum_{j=1}^k \sum_{x \in S_j}^n dist(x, c_j) $$
where $x$ is a data point in cluster $S_j$ and $c_j$ is the cluster center of cluster $S_j$.
In contrast, the Jenks algorithm would minimize $J$:
$$ J = C - \sum_{j=1}^{k-1} dist(c_{j+1}, c_j)$$
Two things to note, however:
- I am really no expert in clustering algorithms, so confirmations, comments, corrections and edits are welcome.
- The source I reference states that $dist()$ computes the Euclidean distance (so, $\sqrt{(d_i - c_j)^2}$), but from everything else I read on K-means it seems that the squared Euclidean distance ($(d_i - c_j)^2$) is what is actually minimized.
Full reference:
Khan, F. (2012). An initial seed selection algorithm for k-means clustering of georeferenced data to improve replicability of cluster assignments for mapping application. Applied Soft Computing Journal, 12(11), 3698–3700. https://doi.org/10.1016/j.asoc.2012.07.021
