Given two ordered sequence $a_1, a_2, \cdots, a_n$ where $a_i \leq a_{i+1}$, and $b_1, b_2, \cdots b_n$ where $b_i \geq b_{i+1}$ and $a_i, b_i \in \mathbb{R}$.
I can construct a new sequence $S$ by merging two sequence $S = (a_1, b_1), \cdots, (a_n, b_n)$.
And the function i'm interested $f: \mathbb{R}^2 \to \mathbb{R}$ has following property: for any $c\in \mathbb{R}$, both $f((x, c)) $ and $f((c, x)) $ are piece-wise linear convex.
Let $s^* = \textbf{argmin}_{s \in S} f(s)$, my question is what are the extra properties does $f$ needs to have, such that $s^* \in \{ s_1, s_n \}$ ?