Abstract
This paper proposes an adaptive primal-dual dynamics for distributed optimization in multi-agent systems. The proposed dynamics incorporates an adaptive synchronization law that reinforces the interconnection strength between the coupled agents. By strengthening the synchronization between the primal variables of the coupled agents, the given law accelerates the convergence of the proposed dynamics to the saddle-point solution. The resulting dynamics is represented as a feedback-interconnected networked system that proves to be passive. The passivity properties of the proposed dynamics are exploited along with the LaSalle's invariance principle for hybrid systems, to establish asymptotic convergence and stability of the saddle-point solution. Further, the primal dynamics is analyzed for the rate of convergence and stronger convergence bounds are established, it is proved that the primal dynamics achieve accelerated convergence under the adaptive synchronization. The robustness of the proposed dynamics is quantified using $L_{2}$ -gain analysis and the correlation between the rate of convergence and robustness of the proposed dynamics are presented. The effectiveness of the proposed dynamics is demonstrated by applying it to solve distributed least squares and distributed support vector machines problems.
Highlights
Distributed optimization remains a subject of substantial research over recent years
Many algorithms are proposed to solve consensus-based distributed optimization problems arising in networked systems, such as the seminal work on distributed subgradient methods [13], distributed primal-dual dynamical
While the adaptive synchronization improves the rate of convergence of the primal-dual dynamics, it simultaneously degrades the robustness of the proposed algorithm wherein the worst-case L2-gain is quantified by γ (< γ ) in (68)
Summary
Distributed optimization remains a subject of substantial research over recent years. The proposed work envelopes the following key points: 1) The proposed algorithm, designated hereafter as the adaptively synchronized distributed primal-dual dynamics (ADPDD), ensures synchronization of the network-wide primal variables to a common trajectory which is driven to the optimal solution. If i and q are neighbors in G with eiq = xi − xq defined as the local synchronization error, the coupling weight can be represented as a function of eiq, i.e. aiq = hi(eiq), where hi : Rl → R monotonically increases in eiq It yields a stronger synchronization between the primal variables of the coupling agents which motivates to incorporate adaptive synchronization to address the convergence rate of the distributed primal-dual dynamics. In both the cases, the Lyapunov function Vk (θjik (τ )) will be non-increasing
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