Doctor of Philosophy
Luis F. Zuluaga
Historically, polynomials are among the most popular class of functions used for empirical modeling in science and engineering. Polynomials are easy to evaluate, appear naturally in many physical (real-world) systems, and can be used to accurately approximate any smooth function. It is not surprising then, that the task of solving polynomial optimization problems; that is, problems where both the objective function and constraints are multivariate polynomials, is ubiquitous and of enormous interest in these fields. Clearly, polynomial op- timization problems encompass a very general class of non-convex optimization problems, including key combinatorial optimization problems. The focus of the first three chapters of this document is to address the solution of polynomial optimization problems in theory and in practice, using a conic optimization approach. Convex optimization has been well studied to solve quadratic constrained quadratic problems. In the first part, convex relaxations for general polynomial optimization problems are discussed. Instead of using the matrix space to study quadratic programs, we study the convex relaxations for POPs through a lifted tensor space, more specifically, using the completely positive tensor cone and the completely positive semidefinite tensor cone. We show that tensor relaxations theoretically yield no-worse global bounds for a class of polynomial optimization problems than relaxation for a QCQP reformulation of the POPs. We also propose an approximation strategy for tensor cones and show empirically the advantage of the tensor relaxation. In the second part, we propose an alternative SDP and SOCP hierarchy to obtain global bounds for general polynomial optimization problems. Comparing with other existing SDP and SOCP hierarchies that uses higher degree sum of square (SOS) polynomials and scaled diagonally sum of square polynomials (SDSOS) when the hierarchy level increases, these proposed hierarchies, using fixed degree SOS and SDSOS polynomials but more of these polynomials, perform numerically better. Numerical results show that the hierarchies we proposed have better performance in terms of tightness of the bound and solution time compared with other hierarchies in the literature. The third chapter deals with Alternating Current Optimal Power Flow problem via a polynomial optimization approach. The Alternating Current Optimal Power Flow (ACOPF) problem is a challenging non-convex optimization problem in power systems. Prior research mainly focuses on using SDP relaxations and SDP-based hierarchies to address the solution of ACOPF problem. In this Chapter, we apply existing SOCP hierarchies to this problem and explore the structure of the network to propose simplified hierarchies for ACOPF problems. Compared with SDP approaches, SOCP approaches are easier to solve and can be used to approximate large scale ACOPF problems. The last chapter also relates to the use of conic optimization techniques, but in this case to pricing in markets with non-convexities. Indeed, it is an application of conic optimization approach to solve a pricing problem in energy systems. Prior research in energy market pricing mainly focus on linear costs in the objective function. Due to the penetration of renewable energies into the current electricity grid, it is important to consider quadratic costs in the objective function, which reflects the ramping costs for traditional generators. This study address the issue how to find the market clearing prices when considering quadratic costs in the objective function.
Kuang, Xiaolong, "Conic Programming Approaches for Polynomial Optimization: Theory and Applications" (2017). Theses and Dissertations. 2952.