Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum 
$$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum 
1. We start by interpreting the given sequence \(a_1, a_2, \ldots, a_{2018}\) where each \(a_i \in \{-1, 1\}\). We need to find the minimal value of the sum \( S = \sum_{1 \leq i < j \leq 2018} a_i a_j \).

2. Consider the sum \( S \) in terms of the total sum of the sequence elements. Let \( k \) be the number of \( 1 \)'s in the sequence and \( 2018 - k \) be the number of \( -1 \)'s. Then, the total sum of the sequence is:
   \[
   \sum_{i=1}^{2018} a_i = k - (2018 - k) = 2k - 2018
   \]

3. The sum \( S \) can be rewritten using the identity for the square of a sum:
   \[
   \left( \sum_{i=1}^{2018} a_i \right)^2 = \sum_{i=1}^{2018} a_i^2 + 2 \sum_{1 \leq i < j \leq 2018} a_i a_j
   \]
   Since \( a_i^2 = 1 \) for all \( i \), we have:
   \[
   (2k - 2018)^2 = 2018 + 2S
   \]

4. Solving for \( S \), we get:
   \[
   2S = (2k - 2018)^2 - 2018
   \]
   \[
   S = \frac{(2k - 2018)^2 - 2018}{2}
   \]

5. To minimize \( S \), we need to minimize \( (2k - 2018)^2 \). The minimum value of \( (2k - 2018)^2 \) is 0, which occurs when \( 2k = 2018 \), or \( k = 1009 \).

6. Substituting \( k = 1009 \) into the expression for \( S \):
   \[
   S = \frac{(2 \cdot 1009 - 2018)^2 - 2018}{2} = \frac{0 - 2018}{2} = -1009
   \]

Therefore, the minimal value of the sum \( S \) is \(-1009\).

The final answer is \(\boxed{-1009}\).