On the Primal-Dual Method of Multipliers and its Applications

Abstract

With ever growing sources of digital data and the reductions in cost of small-scale wireless processing nodes, equipped with various sensors, microprocessors, and communication systems, we are seeing an increasing need for efficient distributed processing algorithms and techniques. This thesis focuses on the Primal-Dual Method of Multipliers (PDMM) as it applies to wireless sensor networks, and develops new algorithms based on PDMM more appropriate for the limitations on processing power, battery life, and memory that these devices suffer from. We develop FSPDMM and QA-PDMM that greatly improve the efficiency of local node computations when dealing with regularized optimization problems and smooth cost function optimization problems, respectively. We combine these approaches to form the FSQA-PDMM algorithm that may be applied to problems with smooth cost functions and non-smooth regularization functions. Additionally, these three methods often eliminate the need for numerical optimization packages, reducing the memory cost on our nodes. We present the FT-PDMM algorithm for finite-time convergence of quadratic consensus problems, reducing the number of in-network iterations required for network convergence. Finally, we present two signal processing applications that benefit from our theoretical work: a distributed sparse near-field acoustic beamformer; and a distributed image fusion algorithm for use in imaging arrays. Simulated experiments confirm the benefit of our approaches, and demonstrate the computational gains to be made by tailoring our techniques towards sensor networks.