Is this problem convex or non-convex? Please motivate your answers.

\begin{aligned} & \underset{\alpha, \ \beta \ \in \mathbb{R} }{\text{minimize}} & & \left\| \mathbf{z} - {\rm Re} \left\{ {\rm diag} \left( \mathbf{A}^H \mathbf{w}\left(\alpha, \beta\right) \mathbf{w}\left(\alpha, \beta\right)^{H} \mathbf{A} \right) \right\} \right\|_2^2 \\ & \text{subject to} & & \underbrace{[\mathbf{w}\left(\alpha, \beta\right)]_k}_{k\textrm{th element of a vector } \mathbf{w}\left(\alpha, \beta\right)} = \underbrace{\exp\left\{ -j \ \alpha \ \left( (k-1)d \right)^{\beta} \right\}}_{{\textrm{complex exponential}}} \ \ \ \forall k = 1,\cdots,m \ \\ &&& \alpha \geq 0 \\ &&& \beta \geq 0 \\ \end{aligned}

The following variables are given/known:

  • column vector $\mathbf{z} \in \mathbb{R}^n$
  • matrix $\mathbf{A} \in \mathbb{C}^{m \times n}$
  • column vector $\mathbf{w}\left(\alpha, \beta\right) \in \mathbb{C}^{m}$ function of $\alpha$ and $\beta$
  • scalar $d \in \mathbb{R}$.

If we can't solve it in closed-form, can this problem be solved numerically at least? Thank you so much in advance.


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