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So, here is my question.

For a SOCP problem with many second order cone constraints, e.g. $m=10000$ in the following formulation, how to solve the problem efficiently?

$$\min{f(x)}$$ $$s.t. ||A_ix+b_i||_2\le c_i^Tx+d_i~~\forall i=1,2,\cdots,m$$ where $x\in R^n$ is the variable and $A_i\in R^{m\times n},~b_i\in R^m,~c_i\in R^n,d_i\in R$ are parameters.

Does solve the corresponding dual problem help improve the computation efficiency?

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    Depends on the algorithm. – 2017-01-12
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    You haven't told us anything about how big $m$ and $n$ are. The sparsity of the $A_{i}$ matrices is also potentially important. – 2017-01-12
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    If you're using an interior-point method, you basically are solving both the primal and the dual simultaneously. In fact, with a symmetric primal dual method, you'll basically approach the solutions to both problems with equal accuracy. – 2017-01-16
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    But the model you have offered here is so general that I would say there's really no reason to expect magic here. A big problem is just going to take a lot of time and memory to solve. – 2017-01-16
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    Thanks a lot, it really takes a lot time for practical problems as I tried... – 2017-01-25

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