[Paper Review] Distributed Convex Optimization with Inequality Constraints over Time-varying Unbalanced Digraphs

Pei Xie, Keyou You+3 more|arXiv (Cornell University)|Dec 29, 2016

Cooperative Communication and Network Coding3 citations

TL;DR

This paper proposes a novel distributed algorithm for convex optimization with inequality constraints over time-varying unbalanced digraphs, using an epigraph reformulation to eliminate unbalancedness effects. It combines distributed subgradient descent with a random-fixed projection method, proving almost sure convergence to a common optimal solution under uniform joint strong connectivity.

ABSTRACT

This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objectives, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast literature on distributed optimization, most of them require the graph to be balanced, which is quite restrictive and not necessary. Very recently, the unbalanced problem has been resolved only for either time-invariant graphs or unconstrained optimization. This work addresses the unbalancedness by focusing on an epigraph form of the constrained optimization. A striking feature is that this novel idea can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.

Motivation & Objective

• To address the lack of distributed algorithms for constrained optimization over time-varying unbalanced digraphs, which are common in real-world networks.
• To overcome the limitation of existing methods that require balanced graphs or are inapplicable to time-varying topologies.
• To design a simple, scalable algorithm that ensures convergence to a common optimal solution without requiring global knowledge or balanced communication matrices.
• To extend the random-fixed projected algorithm to time-varying graphs while preserving feasibility and convergence.

Proposed method

• Reformulate the original constrained optimization problem into an epigraph form to linearize the objective and decouple local objectives, enabling easier handling of unbalanced digraphs.
• Use a two-step iterative algorithm: first, apply distributed subgradient descent on the epigraph form to reach a consensus-like state; second, project each node’s state toward its local constraint set using a distributed version of the Polyak random projection method.
• Introduce an additional projection toward a fixed direction to enhance transient performance and accelerate convergence.
• Employ a time-varying row-stochastic weight matrix to model communication over unbalanced digraphs, ensuring consensus on the average state.
• Use stochastic analysis and Lyapunov-type arguments to prove convergence, leveraging the uniform joint strong connectivity of the graph sequence.
• Apply the push-sum mechanism implicitly through the weight matrix to maintain balance in information exchange despite unbalanced digraphs.

Experimental results

Research questions

• RQ1 Can distributed convex optimization with inequality constraints be solved over time-varying unbalanced digraphs without requiring balanced communication matrices?
• RQ2 How can the unbalancedness of digraphs be effectively managed in a distributed optimization framework?
• RQ3 Can the epigraph reformulation be used to unify local objectives and simplify the design of distributed algorithms for constrained problems?
• RQ4 What conditions on the time-varying graph topology ensure convergence of the distributed algorithm to a common optimal solution?
• RQ5 How can the convergence rate and transient behavior be improved without increasing computational complexity?

Key findings

• The proposed algorithm ensures almost sure convergence of all nodes to a common optimal solution under the assumption of uniform joint strong connectivity of the time-varying digraphs.
• The epigraph reformulation successfully eliminates the impact of unbalanced communication weights by transforming the problem into a linear objective form.
• The convergence of the iterates is proven via stochastic Lyapunov analysis, showing that the expected squared distance to the optimal solution converges almost surely.
• The algorithm maintains feasibility of iterates through a distributed random-fixed projection step, ensuring that each node’s state remains within its local constraint set.
• The transient performance is improved by an additional projection toward a fixed direction, which accelerates convergence without altering the final solution.
• The algorithm is scalable and does not require global knowledge of the network, making it suitable for large-scale distributed systems with dynamic topologies.