Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

TL;DR

This paper provides a comprehensive overview of conic optimization, highlighting recent theoretical advances, convex relaxation techniques for nonconvex problems, and numerical algorithms for large-scale applications, emphasizing its significance across control and machine learning.

Contribution

It offers a detailed survey of conic optimization theory, including convex relaxation hierarchies and scalable algorithms, advancing understanding in control and machine learning contexts.

Findings

Hierarchies of convex relaxations effectively approximate nonconvex problems.

Numerical algorithms enable large-scale conic optimization solutions.

Conic optimization plays a crucial role in control and machine learning applications.

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.