This thesis explores a particular class of distributed optimization methods for various separable resource allocation problems, which are of high interest in a wide array of multi-agent settings. A distinctly motivating application for this thesis is real-time power dispatch of distributed energy resources for providing frequency control in a distribution grid or microgrid with high renewable energy penetration. In this application, it is paramount that agent data be shared as sparsely as possible in the interest of conserving user privacy, and it is required that algorithms scale gracefully as the network size increases to the order of thousands or millions of resources and devices. Distributed algorithms are naturally well-poised to address these challenges, in contrast to more traditional centralized algorithms which scale poorly and require global access to information.

The class of distributed optimization methods explored here can be broadly described as Newton-like or second-order, implying utilization of second-derivative information of the cost functions, in contrast to well-studied gradient-based or first-order methods. We consider three formulations of separable resource-allocation problems and develop a Newton-like algorithm for each. First, the cost function is given by the sum of local agent costs, supplemented with individual linear box constraints and a global matching-constraint in which the sum of agent states must equal a prescribed constant. Second, we consider a stochastic, nested scenario, in which batches of realizations of problems of the first type must be used to gradually learn the optimal value of a parameter which is coupled with the agent costs. Third, we further constrain the agent states to be binary, and we embed the global matching-constraint as a squared penalty term in the cost. The analysis and simulation studies in the subsequent chapters demonstrate the advantages of our approaches over existing methods; most commonly, we note that convergence rates are substantially improved. We supplement our algorithm development for these three problem formulations with a network design technique, in which we can construct a maximally-connected network by adding some edges to the underlying communication graph, and a real demonstration of distributed algorithms on a large set of heterogeneous devices on the UC San Diego microgrid.