In a course I am taking on programming and algorithms for data scientists, I came across the following example of using co-ordinate descent to find a local minimum of a function. However, it is unclear to me where the following two recurrence relations they derive come from.
Question: Find a local minimum of $f(x,y) = x^2 - cxy + y^2$ (where $c \in [0,2)$) using co-ordinate descent.
Recurrence Relations: We start at the arbitrary point $(x_0, y_0)$. The textbook then produces the following two equations (for $i = 1,2, ...$): $$ x_i = \frac{c}2 \cdot y_{i-1} \space \space \text{ and } \space \space y_i = \frac{c}2 \cdot x_i $$
Perhaps, there is something that I missing here (since there are no intermediary steps), but it isn't clear to me where the two recurrence relations come from and would be grateful for any help.
I know that I could find a local minimum through standard differentiation techniques, however, that doesn't help with my understanding the co-ordinate descent algorithmic approach (where deriving the recurrence relations is a requirement).
Assume we are at some arbitrary point in the iteration process $(x_{i-1}, y_{i-1})$. This will allow us to find recurrence relations to find $(x_i,y_i)$ for the next iteration of the process in the form presented in the body of the question.
If we fix $y=y_{i-1}$, then we get $g(x) := f(x,y_{i-1}) = x^2 - (cy_{i-1})x + y_{i-1}^2$ which is a quadratic function of $x$. We now set the next iteration value of $x$ as the minimum of this quadratic, so $x_{i} = \frac{c}{2}y_{i-1}$ (which is a constant since $y_{i-1}$ is known).
Next, we need to find $y_{i}$ so that we can continue to iterate. We do this by fixing the value of $x$ that we have found, and finding the minimal choice of $y$. We define $h(y) := f(x_{i}, y) = x_{i}^2 - (cx_{i})y + y^2$ where $x_{i}$ is a constant (found in the above paragraph). The minimum choice of $y$ here will define our next choice of $y$ in the iteration process so $y_i = \frac{c}2 x_i$ - since $h(y)$ is a quadratic function of $y$.
This gives us the two required recurrence inequalities as stated in the question.
Following up on this, we can then use the two recurrence relations to derive a recurrence relation linking $x_i$ with $x_{i-1}$ and another recurrence relation which relates $y_i$ to $y_{i-1}$ (by using an appropriate substitution).