Consensus-ADMM for General Quadratically Constrained Quadratic Programming

TL;DR

This paper introduces a novel consensus-ADMM algorithm for solving general non-convex quadratically constrained quadratic programming problems, demonstrating improved scalability and performance in applications like beamforming and phase retrieval.

Contribution

The paper presents a reformulation of QCQP into a consensus optimization form allowing efficient ADMM-based solutions with scalable and parallelizable sub-problems.

Findings

Outperforms existing methods in multicast beamforming

Achieves superior results in phase retrieval tasks

Offers scalable and memory-efficient implementation

Abstract

Non-convex quadratically constrained quadratic programming (QCQP) problems have numerous applications in signal processing, machine learning, and wireless communications, albeit the general QCQP is NP-hard, and several interesting special cases are NP-hard as well. This paper proposes a new algorithm for general QCQP. The problem is first reformulated in consensus optimization form, to which the alternating direction method of multipliers (ADMM) can be applied. The reformulation is done in such a way that each of the sub-problems is a QCQP with only one constraint (QCQP-1), which is efficiently solvable irrespective of (non-)convexity. The core components are carefully designed to make the overall algorithm more scalable, including efficient methods for solving QCQP-1, memory efficient implementation, parallel/distributed implementation, and smart initialization. The proposed algorithm is then tested in two applications: multicast beamforming and phase retrieval. The results indicate superior performance over prior state-of-the-art methods.