I created a certain algorithm and now I try to prove it's convergence. Here is the essence of the algorithm:
Given are finite sets $A_i$, positive constants $\alpha_a$, and a sequence $ \vec{\delta}^{(k)} \in \mathbb{R}^n$: $$ A_1,\; \ldots,\; A_n \subset \mathcal{A} $$ $$ B_a = \{i : a \in A_i\}$$ $$ \sum_{a\in \mathcal{A}}\alpha_a = n, \quad 0 < \alpha_a < |B_a| $$ $$\delta_i^{(0)} = 1 \quad i = 1,2,\; \ldots, \;n$$ $$ \beta_a^{(k)} = \sum_{i \in B_a} \delta_i^{(k)} $$ $$ \delta_i^{(k+1)} = \left(\sum_{a\in A_i} \frac{\alpha_a}{\beta_a^{(k)}}\right)^{-1} $$ Prove that $\vec{\delta}^{(k)}$ converges.
I did some numerical experiments but I am not sure that the bounds on $\alpha_a$ are sufficient. You can make stronger assumptions if you want. I'm just lookgin for an idea how to prove this.
Edit: This may help: in experiments this function is always increases: $$ \sum_{a \in A} \alpha_a \log\frac{\alpha_a}{\beta_a} - \sum^n_{i=1} \log \sum_{a\in A_i} \frac{\alpha_a}{\beta_a}$$
Edit 2: You can assume $|A_i| \geq 2 $