Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Abstract

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.

Decomposition Methods for Large-Scale Semidefinite Programs with Chordal Aggregate Sparsity and Partial Orthogonality

“…Semidefinite programs (SDPs) are a type of convex optimization problems that arise in many fields, for example control theory, combinatorics, machine learning, operations research, and fluid dynamics [9,13,17,45]. SDPs generalize other common types of optimization problems such as linear and second-order cone programs [11], and have attracted considerable attention because many nonlinear constraints admit numerically-tractable SDP reformulations or relaxations [41].…”