I solve a nonlinear optimization problem with constraint
$$ \sum_{j=1}^S \ln \Bigg( \Big(1 -x_j c_k + y_j d_k\Big)^2 + \Big( x_j d_k + y_j c_k \Big)^2 \Bigg) \leq 0, \: k = 1, \dots , K $$
where the optimization variables are $\boldsymbol x, \boldsymbol y \in \mathbb R^S$ and the coefficients $\boldsymbol c , \boldsymbol d \in \mathbb R^K$. I derived this constraint from
$$ \prod_{j=1}^S \Big(1 -x_j c_k + y_j d_k\Big)^2 + \Big( x_j d_k + y_j c_k \Big)^2 \leq 1, \: k = 1, \dots , K $$
where the sum of logs seemed more appealing to me than the product of squares.
Does anyone see further simplifications / reformulations of this constraint (e.g. to conic constraints, "less" nonlinear one, ...)?
To be honest, I have not much hope since I think this problem is inherently non-convex for $\boldsymbol x, \boldsymbol y, \boldsymbol c,\boldsymbol d$ taking negative values, but I want to give it a shot, maybe someone sees something.