Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}, \ \ \mathbf{E} \in \mathbb{R}_{+}^{n \times m}$, with $\mathbf{V} \mathbf{1}_{m} = \mathbf{1}_{n}$ and $\mathbf{E}^{T} \mathbf{1}_{n} = \mathbf{1}_{m}$ where $\mathbf{1}_{m}$ is the vector of ones of size $m$.
Define the function $f: \mathbb{R}^m \to \mathbb{R}^m$:
\begin{align} f(\mathbf{q}) = \arg \min_{\mathbf{p}} \left\{ \sum_{j = 1}^m p_j - \sum_{i = 1}^n \left(\sum_{j = 1}^m E_{ij} q_j\right) \log{\left(\min_j \{ \frac{p_j}{V_{ij}} \} \right)} \right\} \end{align}
Define the following iterative process: \begin{align} \mathbf{p}(0) = (\frac{1}{m}, \frac{1}{m}, \dots, \frac{1}{m})^T \\ \mathbf{p}(t) = f(\mathbf{p}(t-1)) \end{align}
My goal is to understand the convergence properties of the above iterative process. That is, I want to understand for which $\mathbf{V}$ and $\mathbf{E}$ this iterative process converges to the fixed point of $f$. I have around 100,000 experiments where I initialize $\mathbf{E}$ and $\mathbf{V}$ randomly and the process always converges. However, I also have the following "hand-crafted" $\mathbf{V}$ and $\mathbf{E}$ for which the process does not converge:
\begin{align} \mathbf{V} = \begin{bmatrix} 1 & 0\\ 0 & 1\\ \end{bmatrix}\\ \mathbf{E} = \begin{bmatrix} 0 & 1\\ 1 & 0\\ \end{bmatrix} \end{align} These experiments seems to suggest to me that for randomly initialized $\mathbf{V}$ and $\mathbf{E}$ the probability of convergence of the iterative process is 1. That is, the probability of getting $\mathbf{V}$ and $\mathbf{E}$ such that the process does not converge is 0.
I want to answer the following questions: Which conditions must $\mathbf{V}$ and $\mathbf{E}$ satisfy for the process to converge.
The most recent approach I took was to use this paper to find the jacobian of $f$ to see for which values of $\mathbf{V}$ and $\mathbf{E}$ the jacobian of the function is less than 1 but the expression I get is too complicated to understand the values of $\mathbf{V}$ and $\mathbf{E}$ for which the process. I am not even sure how to approach the problem and appreciate any pointers.
