Design of Distributed and Robust Optimization Algorithms: A Systems Theoretic Approach

Optimization algorithms are the backbone of many modern technologies. In this thesis, we address the analysis and design of optimization algorithms from a systems theoretic viewpoint. By properly recasting the algorithm design as a controller synthesis problem, we derive methods that enable a systematic design of tailored optimization algorithms. We consider two specific classes of optimization algorithms: (i) distributed, and (ii) robust optimization algorithms. Concerning (i), we utilize ideas from geometric control in an innovative fashion to derive a novel methodology that enables the design of distributed optimization algorithms under minimal assumptions on the graph topology and the structure of the optimization problem. Concerning (ii), we employ robust control techniques to establish a framework for the analysis of existing algorithms as well as the design of novel robust optimization algorithms with specified guarantees.