Given 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 and $\mathbf{p}\in\mathbb{R}^{n_p\times 1}$ and $\mathbf{q}\in\mathbb{R}^{n_q\times 1}$
I am defining $\mathbf{e_p}$ (dim: $n_p \times 1$)
\begin{align} \mathbf{e_p^T} = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} \end{align}
and $\mathbf{e_q}$ (dim: $n_q \times 1$)
\begin{align} \mathbf{e_q^T} = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} \end{align}
$n=n_p+n_q-1$. Moreover $\mathbf{A}\in\mathbb{R}^{n\times n}$ is a positive semi-definite matrix and has an eigen value decomposition $\mathbf{A}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^T$.
FACT: The convolution of $\mathbf{x}\in\mathbb{R}^{n_x \times 1}$ and $\mathbf{y}\in\mathbb{R}^{n_y \times 1}$ is equivalent to a Toeplitz matrix times a vector i-e
\begin{equation} \mathbf{x*y} = \mathbf{Xy}=\mathbf{Yx}=\mathbf{y*x}\tag{2} \end{equation}
where $\mathbf{X}(dim: n\times n_y), \mathbf{Y}(dim: n\times n_x)$ are toeplitz matrices generated from vectors $\mathbf{x}$ and $\mathbf{y}$ respectively. And $n=n_x+n_y-1$
My approach to solve (1) using Alternate Directional Minimization:
- Initialize $\mathbf{q}$ as a random vector with $q_1=1$.
- From $\mathbf{q}$ in step:1, create a toeplitz matrix $\mathbf{Q}\in\mathbb{R}^{n\times n}$ .
- Now solve following for $\mathbf{p}$
\begin{align} \underset{\mathbf{p}}{\text{min}}\hspace{4mm} \mathbf{p}^T \mathbf{Q^TAQ}\mathbf{p} \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0 \tag{3} \end{align}
But $\mathbf{p}$ in (3) has to have a dimension $n\times 1$ instead of $n_p\times 1$ as defined in (1) originally. This is fine as long as last $n_q-1$ entries in $\mathbf{p}$ are zero.
My questions are:
- Is this reformulation of (1) as (3) correct?
- How do I force last $(n_q-1)$ entries in $\mathbf{p}$ to be zero. How should I write it as a constraint.
Update:
For my second question, I think its enough to take a $\mathbf{Z}$ as $n_p \times n_p$ leading sub-matrix of $\mathbf{Q^TAQ}$ and solve
\begin{align} \underset{\mathbf{p}}{\text{min}}\hspace{4mm} \mathbf{p}^T \mathbf{Z}\mathbf{p} \hspace{4mm} \\ s.t \hspace{4mm}\mathbf{p^Te_p}-1=0 \tag{3} \end{align}
Can some one comment on this update?