Conservative Signal Processing Architectures For Asynchronous, Distributed Optimization Part I: General Framework

This paper presents a framework for designing a class of distributed, asynchronous optimization algorithms, realized as signal processing architectures utilizing various conservation principles. The architectures are specifically based on stationarity conditions pertaining to primal and dual variables in a class of generally nonconvex optimization problems. The stationarity conditions, which are closely related to the principles of stationary content and co-content that can be derived using Tellegen's theorem in electrical networks, are in particular transformed via a linear change of coordinates to obtain a set of linear and nonlinear maps that form the basis for implementation. The resulting algorithms specifically operate by processing a linear superposition of primal and dual decision variables using the associated maps, coupled using synchronous or asynchronous delay elements to form a distributed system. A table is provided containing specific example elements that can be assembled to form various optimization algorithms directly from the corresponding problem statements.