I have is an objective function $f : X \to \mathbb{R}$, and $X \subset \mathbb{Z}^2$ and $X$ is a finite discrete set.
My goal is to find the minimum of $f$ with corresponding $x^* \in X$.
Cleary, I can enumerate $X$ and find the goal. But let's suppose there is an alternative, which is partition $X$ into $k$ might overlap subsets $\{ X_1, \cdots, X_i, \cdots, X_k \}$. Namely $X \subseteq \cup_{i=1}^k X_i$ and $X_i \subset X$. And then approximately find minimum $x_i^*$in each $X_i$ and then aggregate for the minimum.
Define $ba_i = \textbf{max}(\{a_j | (a_j, b) \in X_i\})$, $sa_i = \textbf{min}(\{a_j | (a_j, b) \in X_i\})$, $bb_i = \textbf{max}(\{b_j | (a, b_j) \in X_i\})$, $sb_i = \textbf{min}(\{b_j | (a, b_j) \in X_i\})$
Here, each $X_i$ was generated to have this property:
$(ba_i, sb_i) \in X_i, (sa_i, bb_i) \in X_i$
There is two "random" sampling strategy was performed
1, random two sample $x_i, x_j $ from $X_i$ and use the minum as $x_i^*$.
2, pick $(ba_i, sb_i)$ , and $ (sa_i, bb_i)$ use the minum as $x_i^*$.
$f$ is a data-dependent function, I randomly generate some trials. For which $f$ has a high likelihood to be discrete convex. Namely have local minimum equals to global minimum.
From my observation, strategy 2 works way better than 1. But why is that?