Proof for NP-hardness of simultaneous minimization and maximization of a weighted subset
This sounds to me like the Knapsack problem where

$z$ is the weight of each item,
$Z$ is the capacity of the knapsack,
$g$ the value of the item, and
$G$ the value to be achieved.

The problem is NP-complete but solvable, using dynamic programming, in pseudo-polynomial time $O(n \cdot W)$ where $W = \max_{(z,g) \in R}(g)$.
You can prove that the Knapsack Problem is NP-complete by reducing from the Subset sum problem.  Indeed, Subset Sum is a special case of Knapsack where $z = g$; The &quot;weight&quot; of each problem is the same as its &quot;value&quot;.