The following is quoted from "Bertsekas, D. P. (1999). Nonlinear programming (p. 794). Belmont: Athena scientific".
Convergence of Coordinate Descent: Suppose a function $f$ is continuously differtiable, and that for each $x=(x_1,\dots,x_n) \in R^n$ and $i$, $f(x_1,\dots,x_{i-1},\xi,x_{x+1},\dots,x_n)$ viewed as a function of $\xi$, attains a unique minimum $\bar{\xi}$ over $R$, and is monotonically nonincreasing in the interval from $x_i$ to $\bar{\xi}$. Let $\{x^k\}$ be the sequence generated by the coordinate descent method: $x_i^{k+1} \in \arg \min\limits_{\xi \in R} f(x_1^{k+1},\dots,x_{i-1}^{k+1},\xi,x_{i+1}^k,\dots,\dots,x_n^k)$. Then every limit point of $\{x^k\}$ is a stationary point.
Let $g$ be the function $f(x_1,\dots,x_{i-1},\xi,x_{x+1},\dots,x_n)$ w.r.t $\xi$. My question is does $g$ have to attains a unique minimum to make the above proposition work? I have read some literatures and none of them except this book says $g$ has to be a unique minimum, they just say "attains a minimum". If $g$ attains a unique minimum, according to this book Bertsekas, D. P. (1999), $g$ has to be strongly convex and it puts a stricter constraint on $g$.
