		\begin{algorithmic}
			\State initialization: $x_i^0 \in \mathbb{R}^n$ and $z_{i}^0 = 0$
			\For{$t=0, 1, \dots$}
									\vspace{-3ex}
			\State
			\begin{subequations}\label{eq:GTA}
				\begin{align}\label{eq:GTAw} 
				\hspace{-0.3mm} w_i^{t+1} &= w_i^t -\gamma\sum_{j\in \mathcal{N}_i}\!\! {\ell}_{ij} (w_j^{t} - \delta d_j^t) -  \gamma s_i^t + \delta(d_i^{t+1} - d_i^t)
				\\[.5em]%
				s_i^{t+1} &= s_i^t - \! \gamma\!\sum_{j\in \mathcal{N}_i} \ell_{ij} s_j^{t} + \frac{2}{\delta}(f_i(w_i^{t+1})d_i^{t+1} -f_i(w_i^t )d_i^t)\! 
				\end{align}
			\end{subequations}
			\EndFor
		\end{algorithmic}

\begin{algorithmic}
\State initialization: $x_i^0 \in \mathbb{R}^n$ and $z_{i}^0 = 0$
\For{$t=0, 1, \dots$}
\vspace{-3ex}
\State
\begin{subequations}
\begin{align*}
\hspace{-0.3mm} w_i^{t+1} &= w_i^t -\gamma\sum_{j\in \mathcal{N}_i}\!\! {\ell}_{ij} (w_j^{t} - \delta d_j^t) - \gamma s_i^t + \delta(d_i^{t+1} - d_i^t)
\\[.5em]%
s_i^{t+1} &= s_i^t - \! \gamma\!\sum_{j\in \mathcal{N}_i} \ell_{ij} s_j^{t} + \frac{2}{\delta}(f_i(w_i^{t+1})d_i^{t+1} -f_i(w_i^t )d_i^t)\!
\end{align*}
\end{subequations}
\EndFor
\end{algorithmic}
{
    "title": "extremum seeking tracking for derivative-free distributed optimization",
    "id": "2110.04234",
    "abstract": "in this paper, we deal with a network of agents that want to cooperatively minimize the sum of local cost functions depending on a common decision variable. we consider the challenging scenario in which objective functions are unknown and agents have only access to local measurements of their local functions. we propose a novel distributed algorithm that combines a recent gradient tracking policy with an extremum-seeking technique to estimate the global descent direction. the joint use of these two techniques results in a distributed optimization scheme that provides arbitrarily accurate solution estimates through the combination of lyapunov and averaging analysis approaches with consensus theory. we perform numerical simulations in a personalized optimization framework to corroborate the theoretical results.",
    "categories": "math.oc cs.sy eess.sy",
    "doi": "",
    "created": "2021-10-08",
    "updated": "2024-02-06",
    "authors": [
        "nicola mimmo",
        "guido carnevale",
        "andrea testa",
        "giuseppe notarstefano"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2110.04234"
}
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