Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Decentralized Energy Management of Networked Microgrid Based on Alternating-Direction Multiplier Method

With the ever-intensive utilization of distributed generators (DGs) and smart devices, distribution networks are evolving from a hierarchal structure to a distributed structure, which imposes significant challenges to network operators in system dispatch. A distributed energy-management method for a networked microgrid (NM) is proposed to coordinate a large number of DGs for maintaining secure and economic operations in the electricity-market environment. A second-order conic programming model is used to formulate the energy-management problem of an NM. Network decomposition was first carried out, and then a distributed solution for the established optimization model through invoking alternating-direction method of multipliers (ADMM). A modified IEEE 33-bus power system was finally utilized to demonstrate the performance of distributed energy management in an NM

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

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.Comment: 18 page