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Under what condition(s) a unique solution is available for this over-determined system of equations?

$$ x^TA_1x = x^TA_2x = \cdots = x^TA_{N-1}x $$

where $$ A_d = [w^{d(p_i-p_j)}]_{K \times K} \quad, \qquad w = e^{-j \frac{2\pi}N},\quad \{p_1,p_2,\dots,p_K\} \subseteq \{1,2,\dots,N\} , \quad x \in\mathbb{R}_+^K $$

Obviously for this system to be over-determined $ N - 2 > K $ holds.

Also, how to find the least square solution of this quadratic over-determined system of equations?

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