Suppose I have a variant of the knapsack problem:
$$\max_{x} \sum_{i=1}^n v_ix_i$$
$$(\sum_{i=1}^n w_ix_i - W)^2 \leq k$$
for $v_i, w_i \in \mathbb{R}$, $x_i \in \{0,1\}$ and $k \in \mathbb{R}, k > 0$.
Is it known if the decision problem $\sum_{i=1}^n v_ix_i > V$ for some $V$ is NP-Complete?
The problem is clearly in $\mathsf{NP}$. You can see that it is $\mathsf{NP}$-hard by a reduction from subset sum: given a set $X = \{ x_1, \dots, x_m \}$ of positive integers and another integer $T>0$, is there a subset $S$ of $X$ such that $\sum_{x \in S} x = T$?
To reduce to your version of subset sum, create $n=m$ items, where the $i$-th item has value $v_i = 1$ and weight $w_i=2x_i$, and set $W=2T$, $k=1$, $V=0$. There is a solution to this "knapsack" instance if and only if there is a solution to the subset sum problem.
