Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method

Abstract: Various distributed optimization methods have been developed for solving problems which have simple local constraint sets and whose objective function is the sum of local cost functions of distributed agents in a network. Motivated by emerging applications in smart grid and distributed sparse regression, this paper studies distributed optimization methods for solving general problems which have a coupled global cost function and have inequality constraints. We consider a network scenario where each agent has no global knowledge and can access only its local mapping and constraint functions. To solve this problem in a distributed manner, we propose a consensus-based distributed primal-dual perturbation (PDP) algorithm. In the algorithm, agents employ the average consensus technique to estimate the global cost and constraint functions via exchanging messages with neighbors, and meanwhile use a local primal-dual perturbed subgradient method to approach a global optimum. The proposed PDP method not only can handle smooth inequality constraints but also non-smooth constraints such as some sparsity promoting constraints arising in sparse optimization. We prove that the proposed PDP algorithm converges to an optimal primal-dual solution of the original problem, under standard problem and network assumptions. Numerical results illustrating the performance of the proposed algorithm for a distributed demand response control problem in smart grid are also presented.

• The paper presents a consensus-based distributed primal-dual perturbation algorithm that effectively manages globally coupled costs and constraints via local agent communications.
• It integrates primal-dual subgradient updates with consensus mechanisms to ensure convergence under both smooth and non-smooth inequality constraints.
• Numerical results highlight superior convergence and efficiency, underscoring its potential in applications like smart grids and distributed sparse regression.

An Overview of the Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method

The paper "Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method" addresses an important topic in distributed computing, presenting a novel algorithm designed to solve optimization problems with a globally coupled cost function and inequality constraints. This work is particularly significant for applications such as smart grid management and distributed sparse regression, where individual agents must optimize their functions while constrained by a network-wide objective.

Core Contributions

The authors propose the consensus-based distributed primal-dual perturbation (PDP) algorithm for tackling these complex problems in a distributed manner. The algorithm operates over a network where each agent has only local knowledge and can communicate with its neighbors, but not with every node globally. It combines the primal-dual subgradient method with consensus techniques, enabling distributed agents to estimate global cost and constraint functions through iterative message exchanges with neighboring nodes.

Methodology

The convergence of the proposed PDP algorithm is rigorously proven under standard assumptions. This is achieved by utilizing:

1. Primal-Dual Subgradient Updates: The algorithm features updates based upon perturbed points rather than merely relying on the gradients at previous iterates. This perturbation is critical for overcoming inherent non-convexities that may arise due to coupling constraints.
2. Consensus Mechanism: The average consensus method is employed to ensure that all agents can agree on estimates of the global cost and constraints functions, leveraging messages exchanged locally with immediate neighbors.
3. Handling Non-Smooth Constraints: The method's flexibility is highlighted by its capacity to accommodate both smooth and non-smooth inequality constraints, such as those frequently appearing in sparse optimization problems.

Numerical Results and Implications

The practical application of the PDP algorithm is illustrated in the context of a distributed demand response control problem in a smart grid, demonstrating its effectiveness. Numerical solutions exhibit convergence properties that outperform existing distributed primal-dual methods that do not incorporate perturbation techniques. Moreover, the algorithm is computationally more efficient compared to traditional dual decomposition approaches, particularly in the context of large-scale networks.

Theoretical and Practical Implications

This algorithmic approach opens promising avenues for the deployment of efficient distributed systems where direct data sharing among all nodes is impractical due to privacy, computational, or energy constraints. Theoretical implications reinforce the adaptability of primal-dual frameworks when extended with perturbation and consensus mechanisms, suggesting new possibilities for both convex and certain classes of non-convex optimization problems.

Future Directions

While the paper provides substantial advancements in distributed optimization, it also paves the way for future exploration. Potential extensions include scaling the algorithm for larger networks with asynchronous updates or adapting it for more complex network constraints. The adaptability of the PDP algorithm in handling real-time data changes, a critical factor for smart grids and dynamic systems, is another suggested research avenue.

In summary, the proposed distributed consensus-based primal-dual perturbation method represents a robust contribution to solving distributed optimization problems in constrained and coupled environments. This effort enhances both the theoretical understanding and practical feasibility of distributed systems across various fields, marking a significant step towards more autonomous and efficient networked computations.