I am writing a code of the optimization in a this paper.
However, I found that I get a "Inf" for optimal value of this formula below sometimes, so I want to ask, can we get optimal values when we calculate the optimal problem every time? I think we can, but I can't prove this to myself. Does anyone know the explanation?
\begin{array}{rl} \min_{F_k,\rho_k}&\sum_{k=1}^K tr(\mathbf F_k)\\ \text{s.t.}& 0\le\rho_k\le1\\ &\frac{\mathbf h_k^H \mathbf F_k \mathbf h_k}{\bar {r_{k}}}-\sum\limits_{j \neq k}\mathbf h_k^H \mathbf F_j \mathbf h_k \ge \sigma^2_{a_k}+\frac{\sigma^2_{d_k}}{\rho_k}\\ &\mathbf F_k\succeq0\\ &\sum\limits_{j=1}^{K}\mathbf h_k^H \mathbf F_j \mathbf h_k + \sigma^2_{a_k} \ge \frac{\hat P}{1-\rho_k} \end{array}
$\rho_k$ is a power fraction,deciding how much percent of power do the energy harvest,how much percent do information decoding.
$\mathbf h_k$ is a complex circularly symmetric gaussian channel, $CN(0,0.0018)$, as we know, it is random value,every $\mathbf h_k$ is $4$ by $1$ matrix.
$\mathbf F $ is $\mathbf f \mathbf f^H$, and $\mathbf f$ is beamforming, every $\mathbf f$ is $4$ by $1$ matrix.
$\bar {r_k}$ is a SINR value, in here,we set it is 10dB.
$\hat P$ is like a harvested value,but it will change every time when we calculate the optimal value.
$\sigma^2_{d_k}$ and $\sigma^2_{a_k}$ are a noise variance when we are decoding and harvesting energy, they are $-50$dBm and $-70$dBm, is a constant value, not a random I think.
In the code , I write the second and fourth set of constraints as below, in order to obey the DCP rules,or the code can't run
$$\Re\bigg(\frac{\mathbf h_k^H \mathbf F_k \mathbf h_k}{\bar {r_{k}}}-\sum\limits_{j \neq k}\mathbf h_k^H \mathbf F_j \mathbf h_k\bigg) \ge \sigma^2_{a_k}+\frac{\sigma^2_{d_k}}{\rho_k}$$
$$\Re\bigg(\sum\limits_{j=1}^{K}\mathbf h_k^H \mathbf F_j \mathbf h_k + \sigma^2_{a_k}\bigg) \ge \frac{\hat P}{1-\rho_k}$$