Given an SOCP problem
$$ \begin{array}{ll} \text{minimize}&w^Tx\\ \text{subject to} &\|A_i x + b_i\|_2 \le c_i^T x + d_i ~~~~~~~ 1 \le i < N\\ \end{array} $$
where $x$ is partitioned into two groups $x = (x_1, x_2)$, is it possible to write a new optimization problem defined only over $x_2$ such that the solution to this new problem is a subset of the solution to the original problem?