EFIX: Exact Fixed Point Methods for Distributed Optimization

TL;DR

EFIX is a novel distributed optimization method that uses quadratic penalty reformulation and fixed point solvers, achieving exact convergence with competitive efficiency and robustness.

Contribution

The paper introduces EFIX, a new fixed point-based distributed optimization algorithm with proven exact convergence and improved efficiency over existing methods.

Findings

EFIX achieves exact convergence for strongly convex problems.

The method has a worst-case complexity of O(ε^-1).

Numerical results show EFIX is competitive and robust.

Abstract

We consider strongly convex distributed consensus optimization over connected networks. EFIX, the proposed method, is derived using quadratic penalty approach. In more detail, we use the standard reformulation { transforming the original problem into a constrained problem in a higher dimensional space { to define a sequence of suitable quadratic penalty subproblems with increasing penalty parameters. For quadratic objectives, the corresponding sequence consists of quadratic penalty subproblems. For the generic strongly convex case, the objective function is approximated with a quadratic model and hence the sequence of the resulting penalty subproblems is again quadratic. EFIX is then derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver, e.g., Jacobi Over-Relaxation method. The exact convergence is proved as well as the worst case complexity of order O(epsilon^-1) for the quadratic case. In the case of strongly convex generic functions, a standard result for penalty methods is obtained. Numerical results indicate that the method is highly competitive with state-of-the-art exact first order methods, requires smaller computational and communication effort, and is robust to the choice of algorithm parameters.