Argmin on sums of functions

Question

Let $$f(a)=\text{argmin}_{t\in\mathbb{R}} \left\{\sum_{i=1}^n h(a,b_i,t)\right\}$$ I'm wondering if there is any useful result one can derive from the knowledge of the minimizer of each individual terms in the sum, to ie. describe the minimizer of the whole?

Answer 1

An example: let $f(x) = p(x+1)^2, g(x) = q(x-1)^2$, where $p, q$ are constants. Then $f(x) + g(x) = (p+q)x^2 + 2(p-q)x + (p+q)$, whose minimizer occurs at $\frac{p-q}{p+q}$. By varying $p, q$, this minimizer can occur anywhere. But the minimizers of $f,g$ are always at $-1, 1$, respectively.

Of course, in this example you can recover $p, q$ from $f'(-1), g'(1)$. But that is only because we know the form of $f, g$. If all we knew is that they are convex polynomials, there would be no way to determine where the minimum occurs.