optimization equivalence Given the functions $f_1(r,x)$ and $f_2(r,y)$: $[0,1]\times \Bbb R \to \Bbb R ^+$, solve the following problem
$$\underset{r,x,y}{\text{argmin}}\; f_1(r,x)+f_2(r,y) \\
     \text{subject to}\; x^2+y^2=1$$
When is this equivalent to 
$$\underset{r}{\text{argmin}}\; (\underset{x}{\text{argmin}}f_1(r,x)+\underset{y}{\text{argmin}}f_2(r,y)) \\
 \text{subject to}\; x^2+y^2=1\;\;?$$
Generally speaking when do we have
$$\underset{x,y}{\text{argmin}}\; f(x,y)=\underset{x}{\text{argmin}}\; \underset{y}{\text{argmin}}\;f(x,y)\;\;?$$
Would the result be any different if the $f_1$ and $f_2$ were convex?
EDIT:
I believe I need to give mote context to the problem that I am trying to solve.
Simplifying a little bit, I have a statistical model where I am trying to model two groups (parameters 
$x$ and $y$). The parameter $r$ is for both groups. 
I want to minimize the negative log-likelihood with the given constraint. That would correspond to the first problem.
Now in reality, I have more then 50 groups (and the corresponding 50 parameters) and still one "global" parameter $r$.
So the idea (maybe wrong?) is to minimize for a given $r$ without the constraint and then to minimize over $r$ so that the constraint is satisfied. I.e. the second problem should be defined as
$$h(r): r \mapsto (x^*,y^*)=(\underset{x}{\text{argmin}}\; f_1(r,x), \underset{y}{\text{argmin}}\;f_2(r,y)) \\
 \underset{r}{\text{argmin}}\; (||h(r)||_2^2-1)^2$$
Under which conditions this second problem is equivalent to the first one?
 A: Here are two comments. The first comment shows the two approaches you consider are generally not the same,  even if the $f_1(r,x), f_2(r,x)$ functions are convex in $(r,x)$.  The second comment suggests a standard Lagrange approach which may be more helpful. 
Comment 1: An example showing an optimality gap even for convex problems.
Consider the following convex functions: 
\begin{align*}
f_1(x,r) &= (r+x)^2\\
f_2(y,r) &= r + y^2
\end{align*}
Problem 1:
Minimize: $f_1(x,r)+f_2(y,r)$
Subject to:  $(x,y) \in \mathbb{R}^2, x^2+y^2=1, r \in [0,1]$ 
This reduces to minimizing $(r+x)^2 + r + y^2$ subject to the constraints (which in turn is equivalent to minimizing $r^2 + 2rx + r + 1$ subject to $x \in [-1,1], r\in [0,1]$).  It can be shown the optimal solution is $(x^*, y^*,r^*)=(-1,0,1/2)$ and gives an objective value of $f_1(x^*, r^*)+f_2(y^*,r^*)=3/4$. 
Problem 2:
First: For each $r \in [0,1]$ find $\hat{x}(r) = \arg\min_{x\in\mathbb{R}} f_1(x,r)$ and $\hat{y}(r) = \arg\min_{y \in \mathbb{R}} f_2(y,r)$ (I designed the $f_1, f_2$ functions so that the minimizing points are unique).  We get: 
\begin{align*}
\hat{x}(r) &= \arg\min_{x \in \mathbb{R}} [(r+x)^2 ]= -r\\
\hat{y}(r) &= \arg\min_{y \in \mathbb{R}} [r + y^2] = 0 
\end{align*}
Second: Find all $r\in [0,1]$ such that $\hat{x}(r)^2 + \hat{y}(r)^2 = 1$: We want $r^2 + 0^2 = 1$.  The only $r$ that works is $\hat{r}=1$. This gives the vector: 
$$ (\hat{x}, \hat{y}, \hat{r})=(-1, 0, 1) $$
for an objective value of $f_1(\hat{x},\hat{r})+f_2(\hat{y},\hat{r}) = 1$.  This is not as good as the solution $(x^*,y^*,r^*)$ in problem 1 because that solution also satisfies the constraints but gets a better objective value. 
Comment 2: The standard Lagrange multiplier method
It sounds like you have $m$ functions and you want to solve:
Problem A:
Minimize: $\sum_{i=1}^m f_i(x_i,r)$
Subject to: $r \in [0,1], \quad x_i \in \mathbb{R} \quad \forall i \in \{1, ..., m\}, \quad \sum_{i=1}^m x_i^2 = 1$
Let $\lambda$ be a given real number, called a Lagrange multiplier, and consider the simpler problem: 
Problem B:
Minimize: $\sum_{i=1}^m [f_i(x_i,r) + \lambda x_i^2]$
Subject to: $ x_i \in \mathbb{R} \quad \forall i \in \{1, ..., m\}, \quad r \in [0,1]$
Let $(x_i^*(\lambda), r^*(\lambda))$ be a solution to problem B for a given $\lambda$.  If you can find a value $\lambda \in \mathbb{R}$ for which $\sum_{i=1}^m (x_i^*(\lambda))^2=1$, then $(x_i^*(\lambda), r^*(\lambda))$ is also optimal for problem A.  This is true for any  functions $f_i(x_i,r)$, regardless of convexity. I wrote up notes on this general topic, 
see Theorem II.2 in page 7 of these notes: http://ee.usc.edu/stochastic-nets/docs/network-optimization-notes.pdf
Your question also seems similar in spirit to Exercises VIII-F.15-VIII-F.17 in those notes. 
