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

\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.

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.

• Why is the constraint with $\sqrt{t_k}$ a SOCO constraint? – LinAlg Oct 7 '18 at 13:01

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 $$||x|| \leq t$$.
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 $$y$$.