A Fully Parallel Distributed Algorithm for Nonsmooth Convex Optimization With Coupled Constraints: Applications to Distributed Consensus-Based Optimization and Distributed Resource Allocation

This paper aims at collaborative optimization of sum of convex functions over networks subject to globally coupled affine equality and inequality constraints whose partial information is known by each agent. The proposed discrete-time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fully parallel</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed</i> algorithm is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">first</i> of its kind in the sense that it does not require diminishing step size, (sub)gradient, and/or solving a sub-problem at each time step. The algorithm is able to converge to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">optimal</i> solution for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> local convex cost functions (without differentiability or Lipschitz continuity) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> local convex constraint sets (compact or unbounded) of agents with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arbitrary</i> initialization over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> undirected static (non-switching) networks in synchronous protocol. Important applications of the problem can be distributed consensus-based optimization and distributed resource allocation. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">technique</i> utilized here serves as a motivation and guidance for developing several other fully parallel distributed algorithms. Finally, a numerical example of distributed economic dispatch in power systems is provided to demonstrate the efficacy of the results.