Add DAS gradient tracking optimality

src/year2/distributed-autonomous-systems/sections/_optimization.tex

@@ -731,4 +731,25 @@
         % \[
         %     \z_i^{k+1} = \sum_{j=1}^N a_{ij} \z_j^k - (N\alpha) \frac{1}{N} \sum_{j=1}^N \nabla l_k(z_k^k)
         % \]
-\end{description}
+
+        \begin{theorem}[Gradient tracking algorithm optimality] \marginnote{Gradient tracking algorithm optimality}
+            If:
+            \begin{itemize}
+                \item $\matr{A}$ is the adjacency matrix of an undirected and connected communication graph $G$ such that it is doubly stochastic and $a_{ij} > 0$.
+                \item Each cost function $l_i$ is $\mu$-strongly convex and its gradient $L$-Lipschitz continuous.
+            \end{itemize}
+            Then, there exists $\alpha^* > 0$ such that, for any choice of the step size $\alpha \in (0, \alpha^*)$, the sequence of local solutions $\{ \z_i^k \}_{k \in \mathbb{N}}$ of each agent generated by the gradient tracking algorithm asymptotically converges to a consensual optimal solution $\z^*$:
+            \[ \lim_{k \rightarrow \infty} \Vert \z_i^k - \z^* \Vert = 0 \]
+            
+            Moreover, the convergence rate is linear and stability is exponential:
+            \[ 
+                \exists \rho \in (0,1): \Vert \z_i^k - \z^* \Vert \leq \rho \Vert \z_i^{k+1} - \z^* \Vert
+                \,\,\land\,\,
+                \rho \Vert \z_i^{k+1} - \z^* \Vert \leq \rho^k \Vert \z_i^0 - \z^* \Vert
+            \]
+        \end{theorem}
+\end{description}
+
+\begin{remark}
+    It can be shown that gradient tracking also works with non-convex optimization and, under the correct assumptions, converges to a stationary point. 
+\end{remark}