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Primal-Dual Algorithm for Distributed Constrained Optimization
Jinlong Lei, Han-Fu Chen, Hai-Tao Fang
TL;DR
Networked agents must minimize the sum of private local objectives subject to the intersection of their local constraint sets. The paper proposes a constant-step-size distributed primal-dual algorithm based on an augmented Lagrangian; with an appropriate step size, agents reach consensus at an optimum, while the unconstrained time-averaged cost converges at O(1/k).
Problem
The paper studies how networked agents can minimize summed local objectives subject to a global constraint formed by intersecting their private local constraint sets.
Source: §1 · p. 1Method
A constant-step-size distributed primal-dual algorithm uses augmented-Lagrangian saddle-point gradients and local and neighboring-agent information.
Source: §1 · p. 1, §3.2 · p. 3Results
Takeaways & Limitations
Two numerical examples demonstrate the proposed algorithm’s effectiveness for the distributed constrained optimization problem.
Source: §6 · p. 8The consensus guarantee requires a sufficiently small step size depending on initial distance and local cost structure; globally Lipschitz gradients permit a bound independent of initial values.
Source: §4.1 · p. 4
Abstract
from arXiv · showhide
The paper studies a distributed constrained optimization problem, where multiple agents connected in a network collectively minimize the sum of individual objective functions subject to a global constraint being an intersection of the local constraint sets assigned to the agents. Based on the augmented Lagrange method, a distributed primal-dual algorithm with a projection operation included is proposed to solve the problem. It is shown that with appropriately chosen constant step size, the local estimates derived at all agents asymptotically reach a consensus at an optimal solution. In addition, the value of the cost function at the time-averaged estimate converges with rate $O(\frac{1}{k})$ to the optimal value for the unconstrained problem. By these properties the proposed primal-dual algorithm is distinguished from the existing algorithms for distributed constrained optimization. The theoretical analysis is justified by numerical simulations.
1. Introduction
The paper formulates distributed constrained optimization over networked agents and proposes a constant-step-size primal-dual algorithm. It establishes consensus at an optimal solution and an O(1/k) cost convergence rate for the unconstrained case.
1. Introduction
- The problem is to minimize the sum of agents’ local objective functions subject to a global constraint formed from their local constraint sets.The local objectives and constraint sets are known to their respective agents and cannot be shared.Source: §1 · p. 1
- The proposed constant-step-size primal-dual algorithm is based on gradient search for saddle points of an augmented Lagrange function and uses local and neighboring-agent information.Each agent updates its estimate using local data and information from neighboring agents.Source: §1 · p. 1
- With an appropriately chosen constant step size, agents’ estimates reach consensus at an optimal solution; the time-averaged cost converges at O(1/k) to the unconstrained optimum.The rate applies to the unconstrained problem.Source: §1 · p. 1
- The analysis considers deterministic problems with smooth objectives and states weaker assumptions on convex sets than earlier studies requiring compact sets or a nonempty interior.The paper contrasts its assumptions with and the random-case study.Source: §1 · p. 1
- For unconstrained optimization, the paper presents its O(1/k) rate as new and describes the algorithm as an extension of EXTRA to constrained problems.The related-work discussion contrasts this with methods designed for unconstrained problems or converging only to a neighborhood.Source: §1 · p. 1
2. Preliminaries and Problem Statement
The section establishes the graph, convex-analysis, and problem assumptions used to formulate distributed constrained optimization. Agents communicate over an undirected graph and minimize a sum of smooth convex local objectives over the intersection of closed convex local sets.
2.1. Graph Theory
- An undirected graph represents agent-to-agent communication, and its Laplacian is defined as the degree matrix minus the adjacency matrix.For a connected graph, the Laplacian has a simple zero eigenvalue and all other eigenvalues are positive.Source: §2.1 · p. 2
2.2. Gradient, Projection Operator and Normal Cone
- For a nonempty convex set, projection selects the closest point, and a closed set gives a unique projection.The section also introduces the normal cone at a point in a closed convex set.Source: §2.2 · p. 2
2.3. Problem Formulation
- The optimization problem uses each agent’s smooth convex local objective and closed convex constraint set, with the global feasible set formed from the local sets.The formulation assumes a finite solution, a relative interior point in the global set, a connected undirected graph, and locally Lipschitz gradients.Source: §2.3 · p. 2
3. Algorithm Design
The algorithm design recasts the problem as a consensus-constrained optimization and uses a distributed primal-dual method based on an augmented Lagrangian. The analysis uses a Lyapunov function because the augmented Lagrangian is neither strongly convex nor strongly concave.
3. Algorithm Design
- The design first introduces an equivalent problem and a constant-step-size distributed primal-dual algorithm for solving it.The section states that the equivalent formulation requires a connected undirected graph.Source: §3 · p. 2, §3.1 · p. 3
3.1. An Equivalent Problem
- Under the connected-graph formulation, consensus among the agents’ variables makes a solution of the equivalent problem an optimal solution of the original problem.The primal-dual solutions of the formulations correspond to saddle points of the Lagrange function.Source: §3.1 · p. 3
3.2. Distributed Primal-Dual Algorithm
- The distributed primal-dual updates are a gradient algorithm for finding saddle points of an augmented Lagrangian, and convergence is analyzed with a Lyapunov function.The paper states that standard saddle-point gradient analysis does not apply because the augmented Lagrangian is neither strongly convex in X nor strongly concave in Λ.Source: §3.2 · p. 3
4. Convergence Analysis
The convergence analysis establishes that, under a suitable constant step size, agents’ estimates reach consensus at an optimal solution. It also bounds the convergence rate of the time-averaged cost and the consensus residual.
4.1. Main Results
- With 0 < α ≤ 1/(2κ_n) and α < 3/(2l_r), the local estimates converge to a common optimal solution.Here l_r is the local Lipschitz constant of the gradient near the optimal solution.Source: §4.1 · p. 3
4.3. Proof of Theorem 4.3
- The time-averaged estimate’s cost converges to the optimum at O(1/k), while (L ⊗ I_m)X̄_k converges to zero at O(1/k).Source: §4.1 · p. 4
4.2. Proof of Theorem 4.1
- The proof keeps d_k ≤ r and makes V(X_k, Λ_{k+1}) decrease monotonically, then establishes convergence to an optimal solution.The limit is identified as an optimum, and the iterates converge to that limit.Source: §4.2 · pp. 5–6
5. Numerical Simulations
The simulations illustrate the algorithm’s behavior on constrained and unconstrained problems, including comparisons with existing methods. In the unconstrained comparison, the proposed algorithm, EXTRA, and D-NG converge to the optimum, while DGD’s behavior depends on its step-size schedule.
Example 5.1
- In the constrained example, all agents’ estimates converge to the same optimal solution.The example uses different local constraint sets, including noncompact sets, and cost gradients that are only locally Lipschitz continuous.Source: §5 · p. 7
Example 5.2
- The unconstrained comparison uses a 10-agent undirected network with average degree 4 and compares the proposed method with DGD, EXTRA, and D-NG.DGD is tested with constant and two diminishing step-size schedules; the network topology is shown in Figure 2.Source: §5 · pp. 7–8, Figure 2 · p. 7
- The proposed algorithm, EXTRA, and D-NG converge to the optimum; decreasing-step DGD is slowest, while constant-step DGD approaches an optimal-solution neighborhood.Figure 3 plots iteration k against log10(e_k), the normalized relative error.Source: §5 · p. 8, Figure 3 · p. 8
6. Conclusion
The paper proposes a constant-step-size distributed primal-dual algorithm for constrained optimization over a multi-agent network. The agents’ estimates converge to a shared optimum, and in the unconstrained case the time-averaged cost converges at rate O(1/k) to the unconstrained optimal value.
6. Conclusion
- The proposed distributed primal-dual algorithm uses a constant step size to minimize summed local costs subject to the intersection of agents’ local constraints.The conclusion reports convergence for the agents’ estimates and O(1/k) convergence for the unconstrained time-averaged cost.Source: §6 · p. 8
- With a constant step size, all agents’ estimates converge to the same optimal solution; without constraints, the time-averaged cost converges at O(1/k) to the unconstrained optimum.The paper reports numerical support from two examples.Source: §6 · p. 8
- The paper identifies stochastic optimization as a direction for future work, including whether the deterministic case’s desired properties persist.Source: §6 · p. 8