This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^2$, $i\in\mathcal{N}$. The total function is defined as \begin{align} f(r)=\sum_{i=1}^nf_i(r),~~r\in\mathbb{R}^2. \end{align} We know that the local functions $f_i(x)$ and total function $f(r)$ have their own unique optimal solutions defined as $x_i^*$ and $r^*$, respectively.
Intuitively, I think these optimal solutions should satisfy the following relationship
1. if $x_i^*$ are all bounded, then $r^*$ is bounded,
2. or even $r^*$ should be in the convex hull of $x_i^*$.
But I don't know how to prove it. Can someone help me out?
