For a SOCP with many second-order cone constraints, e.g., $m = 10000$ in the following formulation, how to solve the problem efficiently?
$$\min_{x \in \Bbb R^n} f(x) \quad \text{s.t.} \quad \| A_i x + b_i \|_2 \le c_i^T x + d_i, \quad \forall i=1,2,\dots,m$$
where $A_i \in \Bbb R^{m \times n}$, $b_i \in \Bbb R^m$, $c_i \in \Bbb R^n$, $d_i \in \Bbb R$ are given. Does solving the corresponding dual problem help improve the computational efficiency?