Distributed Optimization With Projection-Free Dynamics: A Frank-Wolfe Perspective.

Area of Science:

• Optimization Theory
• Distributed Systems
• Control Theory

Background:

• Distributed constrained optimization presents challenges, particularly with large-scale variables, due to computationally expensive projection operations.
• Existing methods often rely on projection-based dynamics, which can be inefficient in high-dimensional scenarios.

Purpose of the Study:

• To propose a novel distributed projection-free dynamics algorithm for solving constrained optimization problems.
• To enable efficient optimization in multiagent networks without requiring projection operations.
• To provide a theoretical analysis and practical implementation of the proposed method.

Main Methods:

• Employing the Frank-Wolfe method (conditional gradient) to find feasible descent directions via linear suboptimization.
• Designing dynamics for multiagent networks with weight-balanced digraphs to achieve consensus and gradient tracking.
• Developing both continuous-time dynamical systems and a discrete-time scheme.

Main Results:

• Rigorous convergence analysis for the continuous-time dynamical systems.
• Derivation of a discrete-time scheme with a proven convergence rate of O(1/k).
• Demonstration of advantages over existing distributed projection-based and Frank-Wolfe algorithms.

Conclusions:

• The proposed distributed projection-free dynamics offer an efficient alternative for large-scale constrained optimization in multiagent systems.
• The method effectively handles consensus and gradient tracking without computationally intensive projections.
• The theoretical convergence guarantees and practical comparisons highlight the algorithm's efficacy.