EFIX: Exact fixed point methods for distributed optimization

Abstract

AbstractWe 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 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 $${{\mathcal {O}}}(\epsilon ^{-1})$$ O ( ϵ - 1 ) for the quadratic case. In the case of strongly convex generic functions, the 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.