Adnan Nasir

ORCID Identifier(s)

0000-0001-5054-4557

Graduation Semester and Year

2022

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Electrical Engineering

Department

Electrical Engineering

First Advisor

Ramtin Madani

Abstract

This document is prepared to highlight the difference between two very important fields in the realm of Convex Optimization, Machine Learning, Artificial Intelligence and many such sprouting advance fields currently impacting the whole world. The main aim of the recent SDP algorithms is to relax the NP-Hard Convex optimization problems and transform them into amenable jobs without strong theoretical guarantees, but these algorithms tends to approximate the practical in demand applications in many of the aforementioned fields. Moreover, The lack of theoretical guarantee makes the SDP relaxation in-exact and unoptimizable in finite duration. The actual world applications most of the time desire real time optimized results, for example, in applications related to stock exchange, critical space explorations and launches, pattern and feature detection, in cyber security and many more. Many efforts have been made to develop algorithms that can relax the SDPs, and in the same time can reach to a near optimal solution without wasting a lot of time. Many such efforts from the past and including some recently proposed algorithms are discussed in this survey, highlighting important methods used by these prominent researchers such as low rank First and Second order relaxations, with and without sparsity. Due to the demand of making these algorithms work fast on huge data sets, it is increasingly difficult to converge to optimality for any of the contemporary SDP algorithms in finite time. We have proposed a novel SDP algorithm, that utilizes Sequential Parabolic Relaxation to further relax the non-convex constraints. The proposed algorithm not only optimizes the SDP in hand, reaching the optimal solution in a finite amount of time, it also gives a theoretical guarantee of accuracy. We have discussed the proposed algorithm emphasizing the analytical and experimental results on the System Identification and Control, and comparing its performance with state-of-the-art algorithms. After explaining the novel convex relaxation to solve the equations governing the behavior of linear and nonlinear dynamical systems. The proposed relaxation is used for the purpose of system identification. We demonstrate significant improvements in the overall scalability in comparison with the state-of-the-art convex relaxations. For cases where the proposed relaxation is inexact, a sequential algorithm is provided which converges within a finite number of rounds. Theoretical guarantees and simulation results are presented to demonstrate the effectiveness of the proposed approach. This review will also state the mathematical proof ensuring the convergence of the algorithm in finite time. At the end, we tried to pin-point new directions and the challenges that are still ahead in this vast field of SDP optimization.

Keywords

Semidefinite programming, Convex optimization

Disciplines

Electrical and Computer Engineering | Engineering

Comments

Degree granted by The University of Texas at Arlington

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