Definition of local minimum in k-means algorithm I know what a local minimum for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is. The error function in a k-means algorithm gets a vector of assignments and a vector of centers. How does the term local minimum make sense for the error function?

Given a set of observations $X_1, X_2, \ldots, X_N$, where each observation $X_j \in \mathcal R^n$, the $k$-means clustering algorithm's goal is to partition the $N$ observations into $k (\leq N)$ clusters $C = \{C_1, C_2, \ldots, C_k\}$ in order to minimize the within-cluster sum of squares.
The objective function for k-means clustering is
$\underset{\mu}{\mathrm{min}}\;\underset{C}{\mathrm{min}}\sum\limits_{i=1}^{k}\sum\limits_{X_j \in C_i}\|X_j-\mu_i\|^2$
where $\mu=(\mu_1, \mu_2, \ldots \mu_k)$ are cluster centroids for the clusters $C_1,C_2,\ldots,C_k$, respectively.
This function is highly non-convex and potentially prone to have multiple local minima.
The problem is NP-hard and the k-means is a coordinate descent algorithm that fixes one of $\mu$ and $C$  and optimizes the other one.
