## 4 Models and Algorithm Design
### Algorithm Development
#### 4.4.2 Convergence Analysis  
In this section, the analysis of convergence for the algorithm will be conducted. Before delving into the convergence analysis, it is essential to establish specific facts about the directed graph \(\mathcal{G}\) and the weight matrix \(W\). The subsequent Lemmas and Theorems will be stated generically using the index \(i\), ensuring their applicability to all source types. **Lemma 1**. : _Definitions 1 (Equation (25)) and 2 (Equation (28)) hold for the graph \(\mathcal{G}\) and the matrix \(W\). Parameter \(\pi_{i}\) is left normalized eigenvalues vector of the matrix \(W\). Then there exists a parameter \(\beta\leq 0\) and \(\xi\in(0,1)\) such that \(\forall i,j\in\mathcal{G}\) and \(k\geq 0\), it holds that [63]:_  
\[\left|[W^{k}]_{ji}-\pi_{i}\right|\leq\beta\xi^{k},\quad|y_{ii}(k)-\pi_{i}|\leq \beta\xi^{k}.\]  
_Furthermore, a constant \(h>1\) will be introduced such that:_  
\[h^{-1}\leq y_{ii}(k)\leq 1,\quad\forall i,j\in\mathcal{G},\quad k\geq 0.\]  
_This Lemma will subsequently be used to establish results for asymptotic consensus._  
**Lemma 2**. : _In this Lemma, the following sequence will be introduced [63]:_  
\[\theta_{i}(k+1)=\sum_{j=1}^{N}w_{ij}\theta_{j}(k)+\epsilon_{i}(k).\]  
_Here,_ **Definition 1** _still holds for \(w_{ij}\). Subsequently, a variable \(\overline{\theta}(k)\) will be introduced:_  
\[\overline{\theta}(k)=\sum_{i=1}^{N}\pi_{i}\theta_{i}(k).\]  
_where the parameter \(\pi_{i}\) was defined in **Lemma 1**. Furthermore, if it is considered that \(\lim_{k\rightarrow+\infty}||\epsilon_{i}(k)||=0\) then:_  
\[\lim_{k\rightarrow+\infty}||\theta_{i}(k)-\overline{\theta}(k)||=0,\quad \forall i\in\mathcal{G}.\]  
_In this Lemma, the deterministic counterpart to the supermartingale convergence Theorem is thus relied upon to achieve asymptotic convergence for the dual variable \(\lambda_{i}(k)\). ([PERSON]'s martingale convergence Theorems, named after mathematician [PERSON], deal with the limits of supermartingales. Supermartingales are like decreasing sequences in the world of random variables. The main idea is that, under certain conditions, a supermartingale will always converge. This Theorem is akin to the monotone convergence Theorem, which tells us that bounded decreasing sequences converge. )_  
**Lemma 3**. : _In the next step, a non-negative scalar sequence \(\{s(k)\}\) will be introduced for \(\forall k\geq 0\) such that it satisfies [64]:_  
\[s(k+1)\leq\big{(}1+v(k)\big{)}\cdot s(k)-b(k)+c(k), \tag{30}\]  
_where for the parameters \(v(k),b(k),c(k)\geq 0\) with the condition for \(v(k)\) and \(c(k)\):_  
\[\sum_{k=0}^{\infty}v(k)<\infty,\quad\sum_{k=0}^{\infty}c(k)<\infty.\]_It follows that \(\lim_{k\rightarrow+\infty}s(k)=\sigma\), where \(\sigma\) is a non-negative constant for which \(\sigma\geq 0\) and at the same time \(\sum_{k=0}^{\infty}b(k)<\infty\)._  
**Lemma 4**. : **(Asymptotic consensus)** _Here, it will be returned to the algorithm defined by the Equations (22)-(24). Furthermore, it will be assumed that **Assumptions 1-3** and **Definitions 1 and 2** still hold true. Subsequently, a variable \(\overline{\lambda}(k)\) will be introduced [8]:_  
\[\overline{\lambda}(k)=\sum_{i=1}^{N}\pi_{i}\lambda_{i}(k).\]  
_Then the form for the limit can be written as:_  
\[\lim_{k\rightarrow+\infty}||\lambda(k)-\overline{\lambda}(k)||=0,\quad\text {for any }i\in\mathcal{G}.\]  
_This results in the convergence of \(\overline{\lambda}(k)\) to the optimal value reached by all agents in the network. Proof can be found in Appendix A._  
**Lemma 5**. : _It will be assumed that **Assumptions 1-3** and **Definitions 1 and 2** hold true again. Consequently, for any \(u\in R\) and \(k\geq 0\), the following holds [8]:_  
\[