Convergence Analysis of Consensus-ADMM for General QCQP

TL;DR

This paper investigates the convergence behavior of the consensus-ADMM algorithm when applied to general QCQPs, establishing conditions for convergence and demonstrating boundedness and monotonicity of the augmented Lagrangian.

Contribution

It provides a theoretical analysis of the convergence properties of consensus-ADMM for QCQPs, including conditions on parameters and matrix definiteness.

Findings

Augmented Lagrangian decreases monotonically with large enough parameters.

Boundedness of the augmented Lagrangian when the quadratic matrix is positive definite.

Consensus-ADMM converges under specified conditions.

Abstract

We analyze the convergence properties of the consensus-alternating direction method of multipliers (ADMM) for solving general quadratically constrained quadratic programs. We prove that the augmented Lagrangian function value is monotonically non-increasing as long as the augmented Lagrangian parameter is chosen to be sufficiently large. Simulation results show that the augmented Lagrangian function is bounded from below when the matrix in the quadratic term of the objective function is positive definite. In such a case, the consensus-ADMM is convergent.