Title: Distributed and Large-Scale Optimization
Description: This dissertation is motivated by the pressing need for solving real-world large-scale optimization problems with the main objective of developing scalable algorithms that are capable of solving such problems efficiently. Large-scale optimization problems naturally appear in complex systems such as power networks and distributed control systems, which are the main systems of interest in this work. This dissertation aims to address four problems with regards to the theory and application of large-scale optimization problems, which are explained below: Chapter 2: In this chapter, a fast and parallelizable algorithm is developed for an arbitrary decomposable semidefinite program (SDP). Based on the alternating direction method of multipliers, we design a numerical algorithm that has a guaranteed convergence under very mild assumptions. We show that each iteration of this algorithm has a simple closed-form solution, consisting of matrix multiplications and eigenvalue decompositions performed by individual agents as well as information exchanges between neighboring agents.

The cheap iterations of the proposed algorithm enable solving a wide spectrum of real-world large-scale conic optimization problems that could be reformulated as SDP. Chapter 3: Motivated by the application of sparse SDPs to power networks, the objective of this chapter is to design a fast and parallelizable algorithm for solving the SDP relaxation of a large-scale optimal power flow (OPF) problem. OPF is fundamental problem used for the operation and planning of power networks, which is non-convex and NP-hard in the worst case. The proposed algorithm would enable a real-time power network management and improve the system's reliability. In particular, this algorithm helps with the realization of Smart Grid by allowing to make optimal decisions very fast in response to the stochastic nature of renewable energy. The proposed algorithm is evaluated on IEEE benchmark systems. Chapter 4: The design of an optimal distributed controller using an efficient computational method is one of the most fundamental problems in the area of control systems, which remains as an open problem due to its NP-hardness in the worst case.

In this chapter, we first study the infinite-horizon optimal distributed control (ODC) problem (for deterministic systems) and then generalize the results to a stochastic ODC problem (for stochastic systems). Our approach rests on formulating each of these problems as a rank-constrained optimization from which an SDP relaxation can be derived. We show that both problems admit sparse SDP relaxations with solutions of rank at most~3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the rank-3 solution may be deployed to retrieve a near-global controller. We also propose computationally cheap SDP relaxation for each problem and then develop effective heuristic methods to recover a near-optimal controller from the low-rank SDP solution. The design of several near-optimal structured controllers with global optimality degrees above 99\% will be demonstrated. Chapter 5: The frequency control problem in power networks aims to control the global frequency of the system within a tight range by adjusting the output of generators in response to the uncertain and stochastic demand.

The intermittent nature of distributed power generation in smart grid makes the traditional decentralized frequency controllers less efficient and demands distributed controllers that are able to deal with the uncertainty in the system introduced by non-dispatchable supplies (such as renewable energy), fluctuating loads, and measurement noise. Motivated by this need, we study the frequency control problem using the results developed in Chapter 4. In particular, we formulate the problem and then conduct a case study on the IEEE 39-Bus New England system. The objective is to design a near-global optimal distributed frequency controller for the New England test system by optimally adjusting the mechanical power input to each generator based on the real-time measurement received from neighboring generators through a user-defined communication topology.