Tracking-ADMM for distributed constraint-coupled optimization

Abstract

We consider constraint-coupled optimization problems in which agents of a network aim to cooperatively minimize the sum of local objective functions subject to individual constraints and a common linear coupling constraint. We propose a novel optimization algorithm that embeds a dynamic average consensus protocol in the parallel Alternating Direction Method of Multipliers (ADMM) to design a fully distributed scheme for the considered set-up. The dynamic average mechanism allows agents to track the time-varying coupling constraint violation (at the current solution estimates). The tracked version of the constraint violation is then used to update local dual variables in a consensus-based scheme mimicking a parallel ADMM step. Under convexity, we prove that all limit points of the agents' primal solution estimates form an optimal solution of the constraint-coupled (primal) problem. The result is proved by means of a Lyapunov-based analysis simultaneously showing consensus of the dual estimates to a dual optimal solution, convergence of the tracking scheme and asymptotic optimality of primal iterates. A numerical study on optimal charging schedule of plug-in electric vehicles corroborates the theoretical results.

Simultaneous Perturbation Stochastic Approximation-Based Consensus for Tracking Under Unknown-But-Bounded Disturbances

“…The following theorem shows the asymptotically efficient upper bound of estimation residuals provided by algorithm (8).…”

“…To date, there exist a number of approaches for the case when functions F i (x) are convex. In particular, the Alternating Direction Method of Multipliers [7], [8], as well as the subgradient method [9], [10] were proposed. For non-convex tasks, the works [11], [12] develop a large class of distributed algorithms based on various "functionalsurrogate units".…”

Distributed Resource Allocation Algorithms for Multi-Operator Cognitive Communication Systems

“…Therefore, following the algorithm in [14], at iteration t, the algorithm consists of two steps: the local step in which operator n computes a minimizer of the following optimization problem…”

“…where Q ∈ R L×M is a parameter of ADMM that gradually enforces the equality constraints (see [14]), ⋅ 2 denotes 2 norm, c > 0 is a constant penalty parameter, and the superscript t determines the iteration number. Also, the local step for the slack variable is performed as:…”

Distributed Dual Gradient Tracking for Resource Allocation in Unbalanced Networks

“…The extension to general convex functions is performed in [16] by adopting the nonnegative surplus method, at the expense of a slower convergence rate. The ADMM-based algorithms are developed in [17], [18], and algorithms that aim to handle communication delay in time-varying networks and perform event-triggered updates are studied in [19] and [20], respectively. We note that all the above-mentioned works [1], [15]- [20] do not provide explicit convergence rate for their algorithms.…”