Push-DIGing Algorithm

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For $k=0,1,2, \cdots$ do

      $\mathbf{u}(k+1)=\mathbf{C}(k)(\mathbf{u}(k)-\alpha \mathbf{y}(k))$

      $\mathbf{v}(k+1)=\mathbf{C}(k) \mathbf{v}(k) ; \mathbf{V}(k+1)=\operatorname{diag}\{\mathbf{v}(k+1)\}$

      $\mathbf{x}(k+1)=(\mathbf{V}(k+1))^{-1} \mathbf{u}(k+1)$

      $\mathbf{y}(k+1)=\mathbf{C}(k) \mathbf{y}(k)+\nabla \mathbf{f}(\mathbf{x}(k+1))-\nabla \mathbf{f}(\mathbf{x}(k))$

end for

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 假设1: B连通性假设

          \mathcal{G}_{\tilde{B}_{\ominus}}^{\mathrm{dir}}\left(t \tilde{B}_{\ominus}\right) \triangleq\left\{\mathcal{V}, \quad \bigcup_{\ell=t \tilde{B}_{\ominus}}^{(t+1) \tilde{B}_{\ominus}-1} \mathcal{A}(\ell)\right\}