This work considers a general class of convex optimization problems over time-varying, multi-agent networks, that naturally arise in many application domains like energy systems and wireless networks and proposes a novel distributed algorithm based on a combination of dual decomposition and proximal minimization.
Abstract:
We consider a general class of convex optimization problems over time-varying, multi-agent networks, that naturally arise in many application domains like energy systems and wireless networks. In particular, we focus on programs with separable objective functions, local (possibly different) constraint sets and a coupling inequality constraint expressed as the non-negativity of the sum of convex functions, each corresponding to one agent. We propose a novel distributed algorithm to deal with such problems based on a combination of dual decomposition and proximal minimization. Our approach is based on an iterative scheme that enables agents to reach consensus with respect to the dual variables, while preserving information privacy. Specifically, agents are not required to disclose information about their local objective and constraint functions, nor to assume knowledge of the coupling constraint. Our analysis can be thought of as a generalization of dual gradient/subgradient algorithms to a distributed set-up. We show convergence of the proposed algorithm to some optimal dual solution of the centralized problem counterpart, while the primal iterates generated by the algorithm converge to the set of optimal primal solutions. A numerical example demonstrating the efficacy of the proposed algorithm is also provided.
TL;DR: In this article, a dual proximal gradient algorithm is proposed to solve a composite optimization problem with coupling constraints in a multi-agent network based on proximal gradients. But the dual problem is defined by the concept of Fenchel conjugate, which allows for asymmetric individual interpretations of the global constraints.
TL;DR: A distributed Lagrangian algorithm is proposed that solves the deterministic tie-line scheduling problem as well as its robust variant (with policy space approximations) and does not need any form of central coordination.
TL;DR: The algorithm is based on singular perturbation theory, dynamic average consensus, and saddle point dynamics methods to tackle the problem in a fully distributed manner and an analysis of the global optimal solution is presented.
TL;DR: In this paper, the singular perturbation, dynamic average consensus, and saddle point dynamics methods are utilized to construct a dynamical system to seek global optimality, and the theoretical guarantee on the optimality of the solutions is provided as it is proved that the constructed system is semi-globally practically asymptotically stable.
TL;DR: In this article, a dual proximal gradient algorithm was proposed to solve a distributed optimization problem with coupling constraints in a multi-agent network, where the cost function of the agents is composed of smooth and possibly non-smooth parts.
TL;DR: It is argued that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas.
TL;DR: The authors' convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy.
TL;DR: In this article, the authors present a distributed algorithm that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity.
TL;DR: A model for asynchronous distributed computation is presented and it is shown that natural asynchronous distributed versions of a large class of deterministic and stochastic gradient-like algorithms retain the desirable convergence properties of their centralized counterparts.
Q1. What contributions have the authors mentioned in the paper "Distributed constrained convex optimization and consensus via dual decomposition and proximal minimization" ?
The authors consider a general class of convex optimization problems over time-varying, multi-agent networks, that naturally arise in many application domains like energy systems and wireless networks. The authors propose a novel distributed algorithm to deal with such problems based on a combination of dual decomposition and proximal minimization. Their approach is based on an iterative scheme that enables agents to reach consensus with respect to the dual variables, while preserving information privacy. The authors show convergence of the proposed algorithm to some optimal dual solution of the centralized problem counterpart, while the primal iterates generated by the algorithm converge to the set of optimal primal solutions. A numerical example demonstrating the efficacy of the proposed algorithm is also provided.
Q2. what is the weight of agent i to the solution of agent j?
Coefficient aij(k) is the weight that agent i attributes to the solution of agent j at iteration k; aij(k) = 0 means that agent j does not send any information to agent i at iteration k.
Q3. What is the vector of ni decision variables of agent i?
Consider a time-varying network of m agents that communicate to solve the following optimization programP : min {xi∈Xi}mi=1 m∑ i=1 fi(xi)subject to: m∑ i=1 gi(xi) ≤ 0, (1)where for each i = 1, . . . ,m, xi ∈ Rni is the vector of ni decision variables of agent i, fi(·) : Rni →
Q4. Why is the objective and constraint functions in P separable?
Notice that, due to the separable structure of the objective and the constraint functions in P ,ϕ(λ) = m∑ i=1 ϕi(λ) = m∑ i=1 min xi∈Xi Li(xi, λ).
Q5. What is the way to define gi?
It is perhaps worth noticing that this setup comprises also equality coupling constraints like∑mi=1 g̃i(xi) = 0. To this purpose it is enough to define gi = [g̃ > i −g̃>i ]>.
Q6. What is the difference between a primal and a dual update step?
Such an auxiliary sequence is referred to as primal recovery procedure and it is often used in dual decomposition methods, since it has better convergence properties compared to {xi(k)}k≥0 [24], [22], [21].
Q7. What is the purpose of this article?
To account for information privacy and facilitate the development of a computationally tractable solution, the authors seek for a distributed strategy.