Primal-dual algorithm for distributed optimization: A dissipativity-based perspective

TL;DR

This work addresses distributed optimization where the global objective $ ilde f(x)= obreak ilde f(x)= obreak extstyleigl( extstyle extstyle ight) obreak obreak obreak= obreak \sum_i f_i(x)$ is convex while each local cost $f_i$ may be nonconvex, over weight-unbalanced digraphs. It recasts the continuous-time primal-dual dynamics as a Lur'e system—a linear network-dependent block in feedback with a static nonlinear gradient map—and analyzes dissipativity properties of the subsystems. The main contributions show the linear part is dissipative with an appropriate supply rate while the nonlinear feedback is non-passive, and that choosing algorithm gains or engineering the communication graph can ensure exponential convergence to the global optimum. This framework yields new insights into how network topology, gains, and cost nonconvexity jointly influence performance, extending the applicability of primal-dual methods in distributed settings.

Abstract

We study a continuous-time primal-dual algorithm for distributed optimization with nonconvex local cost functions over weight-unbalanced digraphs, and analyze its performance from a dissipativity-based perspective. We first reformulate the algorithm as a Lure type system, consisting of a linear subsystem that relies on the communication topology and the algorithm gains, and a static nonlinear gradient feedback. We then show that the linear subsystem is dissipative with respect to a suitable supply rate, while the nonlinear feedback is not passive. Finally, we establish that, by properly selecting the gains or appropriately designing the communication network, this algorithm converges to an equilibrium at an exponential rate, and thus, achieves an optimal solution to the distributed problem. This work provides new insights into the roles of the network topology, algorithm gains, and cost functions in the performance of a distributed algorithm, and complements existing results from a different viewpoint.

Figures (5)

• Figure 1: Error dynamics as feedback interconnection of two subsystems.
• Figure 2: The communication graphs of the five-agent network.
• Figure 3: Comparison of trajectories under different choices of $\alpha$ and $\beta$: (a) $\bm x^\top \bm L \bm x$; (b) $\bm z^\top \bm L \bm z$.
• Figure 4: Given $\alpha = 5$ and $\beta = 1$, trajectories of $f(\bm x)$ under different $\gamma$: (a) $\gamma = 0.1$; (b) $\gamma = 0.5$.
• Figure 5: Trajectory with $\alpha = 5$, $\beta = 1$ and $\gamma = 0.5$ under the network in Fig. \ref{['fig:grah']}(b): (a) $f(\bm x)$; (b) $\log(\Vert \bm x - \bm x^*\Vert)$.

Theorems & Definitions (9)

• Lemma 1
• Remark 1
• Lemma 2
• Remark 2
• Remark 3
• Theorem 1
• Remark 4
• Theorem 2
• Remark 5