Apologies for the simple question. With $\mathbf{x},\mathbf{v} \in \mathbb{R}^n$, minimize $f(\mathbf{x}): \mathbb {R}^n \rightarrow \mathbb{R}$.
\begin{equation}\tag{*} \begin{array}{c} \text{min} \hspace{4mm} \mathbf{x}^T \mathbf{v}\mathbf{v}^T\mathbf{x} \end{array} \end{equation}
My approach: Any $\mathbf{x}^*$ that is orthogonal to $\mathbf{v}$ will make $\mathbf{x}^T \mathbf{v}\mathbf{v}^T\mathbf{x}=0$. Since $\mathbf{x}$ is used later in my bigger problem, I want to get $\mathbf{x}^*$ that has minimum norm among all possible orthogonal vectors.
My question: How should I write the objective function with this minimum norm constraint. Secondly, since $\mathbf{0}$ is orthogonal to all vectors in $\mathbb{R}^n$ and it has a minimum norm, what should I do to avoid trivial solution i-e $\mathbf{x}^*\neq \mathbf{0}$.
First Edit:
How about reformulating in the following form
\begin{equation}\tag{**} \begin{array}{c} \text{min} \hspace{4mm} \|\mathbf{x}\|_2 \\ s.t. \mathbf{x}^T\mathbf{v}=0 \end{array} \end{equation}
Second Edit:
\begin{align*}\tag{***} \begin{array}{c} \text{min} \hspace{4mm} \|\mathbf{x}\|_2 \\ s.t. \mathbf{A}^T\mathbf{x}=\mathbf{b} \end{array} \end{align*}
where $\mathbf{e}\in \mathbb{R}^n$
\begin{align*} \mathbf{e}= \begin{bmatrix} 1 & 0 &\cdots & 0 \end{bmatrix}^T \end{align*}
\begin{align*} \mathbf{A}^T= \begin{bmatrix} \mathbf{v}^T \\ \mathbf{e}^T \end{bmatrix} \end{align*}
\begin{align*} \mathbf{b}= \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{align*}