Let $\mathbf{Y}$ be a square matrix, $\mathbf{Y} \in \mathbb{R}^{n \times n} $, that can be obtained via a nonlinear operator, $f$, with respect to $\mathbf{x}$, i.e. $\mathbf Y \left(\mathbf{x} \right) =f \left( \mathbf{x} \right)$. And let $\mathbf{x}$ be a vector, $\mathbf{x} \in \mathbb{R}^{n} $, containing either $1's$ or $0's$, given by the following Heaviside function:
$$ \mathbf{x}=h( \mathbf{\tilde{Y}_{i,i}} - \alpha)=\begin{cases} 0~~if~~ \mathbf{\tilde{Y}_{i,i}} \lt \alpha \\ 1~~if~~ \mathbf{\tilde{Y}_{i,i}} \geq \alpha \\ \end{cases} $$ where $\alpha$ is some scalar constant; $\mathbf{\tilde{Y}_{i,i}}$ are the diagonal elements of $\mathbf{\tilde{Y}}$, where $\mathbf{\tilde{Y}}$ is simply the matrix $\mathbf{{Y}}$ from a previous iteration.
Basically, the vector $\mathbf{x}$ is unknown; the "right" value of $\mathbf{x}$ is the one that makes the matrix $\mathbf{{Y}}$ an identity matrix (or close to the identity). Hence, I write the objective function as:
$$ \begin{equation*} \begin{aligned} & \underset{\mathbf{x}^{k+1}}{\text{minimize}} & & \Vert \mathbf{Y}(\mathbf{x}^{k+1}) - \mathbf{I}_{n} \Vert _{2}^{2} \\ & \text{subject to} & & \mathbf{x}^{k+1}=h( \mathbf{\tilde{Y}_{i,i}} - \alpha)=\begin{cases} 0~~if~~ \mathbf{\tilde{Y}_{i,i}} \lt \alpha \\ 1~~if~~ \mathbf{\tilde{Y}_{i,i}} \geq \alpha \\ \end{cases} &, where~ \mathbf{\tilde{Y}}=\mathbf{Y}^{k} \end{aligned} \end{equation*} $$ where $\mathbf{I}_{n}$ is an $n\times n$ identity matrix; $k$ is the current iteration number.
The question is this:
can I solve for $\mathbf{x}$ when $\mathbf{x}$, is, itself, a function? If so,
- Is there a name for such problems?
- How can I write this differently?