If $g(x) = \sum_{i=1}^n g_i(x)$ where each $g_i(x)$ is a strictly concave function with maximum $x_i^* = \operatorname{argmax}\limits_x g_i(x)$, is it true that $$x^* = \operatorname{argmax}\limits_x g(x)$$ must be in the interval $[\min(x_i^*),\max(x_i^*)]$?

I know this is not the case for $g: \mathbb{R}^n \to \mathbb{R}$ from question A property of the minimum of a sum of convex functions

, but is it true for $g: \mathbb{R} \to \mathbb{R}$


Yes. Note that $g$ has a subgradient at $\min(x_i^*)$ that has a nonnegative slope, while there is a subgradient at $\max(x_i^*)$ that has a nonpositive slope. This doesn't even need strict concavity. If the functions are concave, there is an $x_0 \in [\min(x_i^*), \max(x_i^*)]$ so that $g(x_0)$ is the maximum of $g$.

If the functions are all strictly concave and not all $x_i^*$ are identical, the minimum lies in fact in the interior of the interval.


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