Under what conditions on $f$ is the following true?
$$\arg\min_{\alpha_1,\ldots,\alpha_d\in\mathbb{R}^d}\sum_{i=1}^d f(x,\alpha_i)=\{ \arg\min_{\alpha_i\in\mathbb{R}} f(x,\alpha_i)\}_{i=1}^d=\{\hat\alpha_1,\ldots\hat\alpha_d\}$$
But now that I think of it, is it not trivially true if $f$ is a scalar function, defined as piecewise and disjoint on its domain?