Why is this problem (multiuser transmit beamforming problem) an SOCP?

Question

I am reading this article, in which the authors state that the problem

\begin{equation*} \begin{aligned} & \min_{\mathbf w_1 \cdots \mathbf w_K} \sum_{k = 1}^K ||\mathbf w_k||_2^2\\ & \text{s.t.} && \frac{|\mathbf w_k^H\mathbf h_k|^2}{\sum_{l \ne k} |\mathbf w_l^H\mathbf h_k|^2 + \sigma^2_k} \ge c_k,\ \forall k\in \{1, \cdots, K\} \end{aligned} \end{equation*}

, where $\mathbf w_k \in \mathbb C^n$, is an SOCP. I would like to prove this myself, and here's my attempt.

Obviously, we can move the quadratic term in objective function into the constraints by introducing a new variable $\mathbf t \in \mathbb R^K$, and rewrite the problem into:

\begin{equation*} \begin{aligned} & \min_{\mathbf w_1 \cdots \mathbf w_K, \mathbf t} \mathbf 1^T \mathbf t\\ & \text{s.t.} && \frac{|\mathbf w_k^H\mathbf h_k|^2}{\sum_{l \ne k} |\mathbf w_l^H\mathbf h_k|^2 + \sigma^2_k} \ge c_k,\ \forall k\in \{1, \cdots, K\}\\ &&& ||\mathbf w_k||_2 \le \sqrt{t_k},\ \forall k \in \{1, \cdots, K\}. \end{aligned} \end{equation*}

The next step should be to rewrite the other constraint into second-order cone form, but I have no idea how to proceed.

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Update:

I've read this reference(page 24, 18.29) but a few details still bother me.

1) I thought an SOCP must have its objective in the $f^Tx$ form, but in the reference, the objective function is the sum of $d$ second-order terms.

2) Why is it an SOCP with the constraint $Im [\mathbf w_i^H \mathbf h_i] = 0$? I can't find a way to rewrite it to the standard $Fx = g$ form.

Answer 1

Multiply with the denominator to get: $$c_k \left( \sum_{l \ne k} |\mathbf w_l^H\mathbf h_k|^2 + \sigma^2_k\right) \le |\mathbf w_k^H\mathbf h_k|^2$$ This constraint is of the form $||x||^2 \leq |t|^2$, which is not equivalent to $||x|| \leq t$. Your problem in the stated form is not convex and not SOCO.

If you read reference 2, they add the extra constraints that $t$ is nonnegative and that the imaginary part of $t$ is $0$. Under those conditions, $||x||^2 \leq |t|^2$ is equivalent to the SOCO constraint $||x|| \leq t$.

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1. The objective can be written as $\sum t_k$ with $||w_k||^2 \leq t_k$. Use that $x^T x \leq t$ is equal to $||[2x; t-1]|| \leq t+1$.

2. For a complex number $x+yi$, the optimization variables are $x$ and $y$.