My objective function is \begin{align} \underset{\mathbf{p},\mathbf{q}}{\text{min}}\hspace{4mm} (\mathbf{p*q})^T \mathbf{A}(\mathbf{p*q}) \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0\\\mathbf{q^Te_q}-1=0 \tag{1} \end{align}
where $\mathbf{*}$ represents discrete-time convolution (not to be confused with circular convolution) and $\mathbf{p}\in\mathbb{R}^{n_1\times 1}$ and $\mathbf{q}\in\mathbb{R}^{n_2\times 1}$
I am defining $\mathbf{e_p}$ (dim: $n_1 \times 1$)
\begin{align} \mathbf{e_p^T} = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} \end{align}
and $\mathbf{e_q}$ (dim: $n_2 \times 1$)
\begin{align} \mathbf{e_q^T} = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} \end{align}
$n=n_1+n_2-1$. Moreover $\mathbf{A}\in\mathbb{R}^{n\times n}$ is a positive semi-definite matrix and has an eigen value decomposition $\mathbf{A}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^T$.
My question is:
- Is this a convex optimization problem?
- If not, then under what conditions can we get a closed form solution?