I encounter the following optimization problem, but I can't solve it.

Given $N$ variables satisfying $0 <= x_1 < x_2 < x_3 < ... < x_N <= 1$ and an integer $K$ no large than $N$, find the values of $\{x_i\}$ that maximize the following function.

$$\sum_{S \subset \{1,2,..., N\}, |S| = K} \prod_{i<j, i,j \in S} (x_j - x_i)^2.$$

This problem is somehow related to Vandermonde matrix. Each additional term in the above target function is just the square of the determinate of Vandermonde matrix generated by the $K$ selected variables belonging set $S$.