Primal–dual algorithm for distributed constrained optimization

Abstract

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( 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.

Distributed Stochastic Approximation Algorithm With Expanding Truncations

“…This can be seen from the following consideration: i) If σ j,k ≤ σ i,k ∀j ∈ N i (k), then from (6) we deriveσ i,k = σ i,k . Since σ i,k+1 > σ i,k , by (9) it follows that x i,k+1 > Mσ i,k , and hence from (8) we derive x i,k+1 = x * . ii) If there exists j ∈ N i (k) such that σ j,k > σ i,k , then from (6) we deriveσ i,k = max j∈Ni(k) σ j,k > σ i,k , and from (7) we have x i,k+1 = x * .…”

“…By (48) and (49), we obtain σ i,k+1 =σ i,k = σ ∀k ≥ k 0 ∀i ∈ V. Thus, for any k ≥ k 0 and any i ∈ V, we have that x i,k+1 ≤ M σ by (9), and x i,k+1 = x i,k+1 by (8). Then for any i ∈ V, {x i,k } is bounded and (14) follows from (50).…”

“…ii) If there exists j ∈ N i (k) such that σ j,k > σ i,k , then from (6) we deriveσ i,k = max j∈Ni(k) σ j,k > σ i,k , and from (7) we have x i,k+1 = x * . Consequently, by (8) we have x i,k+1 = x * .…”

Distributed Continuous-Time and Discrete-Time Optimization With Nonuniform Unbounded Convex Constraint Sets and Nonuniform Stepsizes

“…Article [1] studied nonuniform convex constraints but the communication graph is complete and all the edge weights are assumed to be equal. Founded on [1], articles [24], [25] gave some results on nonuniform convex constraints but the communication graph is constant and connected, and the objective functions are assumed to be strongly convex or some intermediate variables need be transmitted besides the agent states. Article [29] studied a distributed optimization problem with nonuniform convex constraints and gave conditions to guarantee the optimal convergence of the team objective function, but the subgradients and the convex constraint sets are both bounded.…”

Convergence rate analysis of a subgradient averaging algorithm for distributed optimisation with different constraint sets

“…This latter feature can speed up practical convergence. • We show by means of a counterexample the necessity [11] x [17], [19] [7], [14], [20] [6], [12], [15] x With (sub)gradient averaging [9], [18] [9], [18] x x [8], [13], [16] x our work x of developing a new algorithmic machinery to capture the case of different constraint sets per agent, as a direct adaptation of the algorithm in [8] may fail to converge if agents are subject to different constraint sets. • We show that the iterates generated by our proposed algorithm converge to some minimizer of the centralized problem counterpart for step sizes of the form c(k) = η k+1 , η > 0, while we also establish a convergence rate of O( log k √ k ) for convergence in value and step size of the form c(k) = η √ k+1 .…”