I am interested in solving the following equation in $t$,
\begin{equation} \| \boldsymbol{p} + \boldsymbol{D}^t \boldsymbol{q} \|^2 \approx \epsilon \,. \end{equation}
Here $\boldsymbol{p} \in \mathbb{R}^N$, $\boldsymbol{q} \in \mathbb{R}^N$, and $\boldsymbol{D}$ is a diagonal matrix, $\epsilon>0$ and $t$ is a positive integer.
The $\approx$ means the closest positive integer $t$ that can solve the equation. In other words, the problem can be defined as,
\begin{equation} t^*=\arg\min_{t \in \mathbb{N}} | \| \boldsymbol{p} + \boldsymbol{D}^t \boldsymbol{q} \|^2 - \epsilon |\,. \end{equation}
In scalar notation the problem is,
\begin{equation} t^*=\arg\min_{t \in \mathbb{N}} | \sum_{n=1}^N (p_n + d_n^t q_n)^2 - \epsilon |\,. \end{equation}
I believe a closed form solution does not exist, but I will be happy if I could solve,
\begin{equation} t^*=\arg\min_{t \in \mathbb{N}} | UB[\sum_{n=1}^N (p_n + d_n^t q_n)^2] - \epsilon |\,. \end{equation}
where $UB[.]$ means an upper bound. So the question really is about seeking a reasonably good upper bound on $\sum_{n=1}^N (p_n + d_n^t q_n)^2$ for which we can obtain a closed form expression for $t^*$.
Ideally, I do not want to make assumption about the signs (positive/negative) of $p_n$, $d_n$, and $q_n$.
Any idea would be highly appreciated.
Golabi