I would like the prove the following identity:
$$\min\limits_{\mathbf{x},\mathbf{y}}[f(\mathbf{x})+g(\mathbf{y})]=\min\limits_{\mathbf{x}}[f(\mathbf{x})]+\min\limits_{\mathbf{y}}[g(\mathbf{y})]$$
At first, I supposed that this should be an inequality, because I tried proving this by doing:
\begin{align} \min\limits_{\mathbf{x},\mathbf{y}}[f(\mathbf{x})+g(\mathbf{y})]\geq \min\limits_{\mathbf{x},\mathbf{y}}[f(\mathbf{x})]+\min\limits_{\mathbf{x},\mathbf{y}}[g(\mathbf{y})] \end{align}
But I am not sure if this is correct...
Could someone help me proving this by using some basic principles and properties? Also, are there any conditions/constraints over the functions so that the proposed equality holds?
You have $$\min_{x,y} (f(x)+g(y)) \leq f(x)+g(y), \quad \text{for all}\,\, x,y$$ which implies $$\min_{x,y} (f(x)+g(y)) \leq \min_x f(x)+ \min_y g(y)$$ On the other hand $$f(x)+g(y) \geq \min_x f(x) + \min_y g(y), \quad \text{for all}\,\, x,y$$ therefore $$\min_{x,y} (f(x)+g(y)) \geq \min_x f(x)+ \min_y g(y)$$ So you have the equality.