Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x \tag{1} $$
where $A$ is an $n \times n$ symmetric matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ with $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.$$
To avoid trivial or degenerate solutions, we restrict ourselves to the solutions $x \neq 0$ with $\| x \|=1$; indeed, any solution $x$ of this equation remains a solution after a linear transformation.
Such an equation appears in non-convex optimization, where all of its solutions need to be found so that the problem is fully solved. In fact the original system that needs to be solved is of the form
$$\color{blue}{\left [ \left(A + \text{sgn}(x)\text{sgn}(x)^T \right )x- y \, x \right ] \odot \text{sgn}(x) =0.}, \tag{2} $$
which reduces to the above system if we assume $x_i \neq 0, i=,\dots,n$. Also, if $A$ is a diagonal matrix, the two systems (1) and (2) become the same.
A naïve approach to solve the equation (1) is to fix $\text{sgn}(x)=a$ for any $a \in \{-1,1 \}^n$, and then solve the system. One can see that if $a$ is considered, $-a$ is not needed to be assessed as both produce the same matrix of $aa^T$. Hence, $2^{n-1}$ cases should be examined. After fixing $\text{sgn}(x)=a$, each solution $y$ and $x$ of the resulting system:
$$(A + aa^T )x= y x$$
is nothing but an eigenvalue and its normalized eigenvector of matrix $A+aa^T.$ Such a solution is a correct solution for our equation only if $\text{sgn}(x)=a$ or $\text{sgn}(x)=-a.$
When $A=0$, for each $a$, $\text{rank}(aa^T)=1$, and the system has the following $n$ solutions $(x,y)$:
$$(e_1,1),\dots, (e_n,1).$$
I have two related questions:
How many solutions can this system have (my guess the number of solutions is of order $O(n)$ though $2^{n-1}$ different possibilities can be obtained by fixing $\text{sgn}(x)$)?
Is there any efficient procedure to find all solutions of the above system (the computational complexity of the naïve approach is of exponential order)?
