Is there any relationship between the locations of local maximas of two different functions and the location of the maxima of their sum. That is, if $$\alpha_1 = \arg \max f_1(\alpha)$$ and $$\alpha_2 = \arg \max f_2(\alpha),$$where that $\alpha_1 \ne \alpha_2$. Might there be any function $g(\alpha_1,\alpha_2)$ such that $$\alpha_s = \arg \max (f_1(\alpha) + f_2(\alpha)) = g(\alpha_1,\alpha_2) ?$$
NB: $\alpha \in \mathcal{R}$, $f:\alpha \longrightarrow \mathcal{R}$.
Thank you.
In general, no. Consider the following functions: $$f_1(x)=-2|x|, \: f_2(x) = -2|x - 1|$$ $$g_1(x) = -x^2, \: g_2(x) = -(x-1)^2$$ Their maxima are pretty clear.
Where are the maxima of: $$f_1 + f_2? \quad g_1 + g_2? \quad f_1 + g_2?$$
Note that: $$\arg \max f_1(\alpha) = \arg \max g_1(\alpha) = 0$$ $$\arg \max f_2(\alpha) = \arg \max g_2(\alpha) = 1$$