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Introduction to Convex Optimization

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Introduction to Convex Optimization Lecture Notes: Introduction to Convex Optimization Changho Suh1 June 4, 2019 1Changho Suh is an Associate Professor in the School of Electrical Engineering at Korea Advanced Institute of Science and Technology, South Korea (Email: chsuh@kaist:ac:kr). EE523 Convex Optimization February 26, 2019 KAIST, Spring 2019 Lecture 1: Logistics and Overview About instructor Welcome to EE523: Convex Optimization! My name is Changho Suh, an instructor of the course. A brief introduction of myself. A long time ago, I was one of the students in KAIST like you. I spent six years at KAIST to obtain the Bachelor and Master degrees all from Electrical Engineering in 2000 and 2002, respectively. And then I left academia, joining Samsung electronics. At Samsung, I worked on the design of wireless communication systems like 4G- LTE systems. Spending four and a half years, I then left industry, joining UC-Berkeley where I obtained the PhD degree in 2011. I then joined MIT as a postdoc, spending around one year. And then I came back to KAIST. My research interests include information theory and machine learning which have something to do with the optimization theory, which I am going to cover in part from this course. Today's lecture In today's lecture, we will cover two very basic stuffs. The first is logistics of this course: details as to how the course is organized and will proceed. The second thing to cover is a brief overview to this course. In the second part, I am going to tell you a story of how convex optimization was developed, as well as what we will cover from this course. My contact information, office hours and TAs' information See syllabus uploaded on the course website. One special note: if you cannot make it neither for my office hours nor for TAs' ones, please send me an email to make an appointment in different time slots. Prerequisite The key prerequisite for this course is to have a good background in linear algebra. In terms of a course, this means that you should have taken the following course: MAS109, which is an introductory-level course on linear algebra. This is the one that is offered in the Department of Mathematical Sciences. Some of you might take a different yet equivalent course from other departments. This is also okay. Taking a somewhat advanced-level course (e.g., MAS212: Linear Algebra) is optional although it is recommended. If you feel a bit uncomfortable although you took the relevant course(s), you may want to review a material (part of a linear algebra course note at Stanford) that I uploaded on the course website. Another prerequisite for the course is a familiarity with the concept on probability. In terms of a course, this means that you are expected to be somewhat comfortable with the contents dealt in EE210, which is an undergraduate-level course on probability. Taking an advanced course like EE528 (Random Processes) is optional. This is not a strong prerequisite. In fact, the convex optimization concept itself has nothing to do with probability. But some problems (especially the ones that arise in recent trending fields like machine learning and artificial intelligence, and I will also touch upon a bit in this course) deal with some quantities which are random and therefore are described with probability distributions. This is the only reason that understanding the concept on probability is needed for this course. Hence, reviewing another material (an easy- 1 level probability course note at MIT) that I uploaded on the course website may suffice. There must be a reason as to why linear algebra is crucial for this course. The reason is related to the definition of optimization, which subsumes convex optimization (which we will study throughout this course) as a special case. A somewhat casual definition of optimization is to make the best choice among many candidates (or alternatives). A more math-oriented definition of optimization (which we should rely on as scientists and/or engineers) is to choose an optimization variable (or decision variable) so as to minimize or maximize a certain quantity of interest possibly given some constraint(s). Here the optimization variable and the certain quantity are the ones that relate the optimization to linear algebra. In many situations, the optimization variable are multiple real values which can be succinctly represented as a vector. Also the certain quantity (which is a function of the optimization variable) can be represented as a function that involves matrix-vector multiplication and/or matrix-matrix multiplication, which are basic operations that arise in linear algebra. In fact, many operations and techniques in linear algebra help formulating an optimization problem in a very succinct manner, and therefore help theorizing the optimization field. This is the very reason that this course requires a good understanding and manipulation techniques on linear algebra. Some of you may not be well trained with expressing an interested quantity with vector/matrix forms. Please don't be offended. You will have lots of chances to be trained via some examples that will be covered in lectures and homeworks as the course progresses. Whenever some advanced techniques are needed, then I will provide detailed explanations and/or materials which serve you to understand the required techniques. If you think that you lack these prerequisites, then come and consult with me so that I can help you as much as possible. Course website We have a course website on the KLMS system. You can simply login with your portal ID. If you want to only sit in this course (or cannot register the course for some reason), please let me or one of TAs know your email address. Upon request, we are willing to distribute course materials to the email address that you sent us. Text The organization of the course will follow mostly but in part by: Prof. Laurent El Ghaoui's livebook (LB for short), titled \Optimization Models and Applications". To access, you need to register at http://livebooklabs.com/keeppies/c5a5868ce26b8125. A good news is that the regis- tration comes for free. I am going to provide you with lecture slides (LS for short) which I will use during lectures, as well as course notes (CN for short) like the one that you are now reading. Most times, these materials will be posted at night on a day before class, but sometimes course notes may be uploaded after lectures. These materials are almost self-contained, and cover the entire contents that will show up in homeworks and exams. So if you want to make a minimal effort to this course (for some personal reason), then these materials may suffice to pass the course with a reasonable grade. For those who have enough energy, passion and time, I recommend you to consult with other references: (1) Calafiore & El Ghaoui, \Optimization Models", Cambridge University Press, Oct. 2014; (2) Boyd & Vandenberghe, \Convex Optimization", Cambridge University Press, 2004 (available online at http://stanford.edu/∼boyd/cvxbook/bv cvxbook.pdf). Sometimes, I will make some homework problems from these references. If so, I will let you know and upload a 2 soft copy of the relevant part of the books. Problem sets There will be weekly or bi-weekly problem sets. So there would be seven to eight problem sets in total. Solutions will be usually available at the end of the due date. This means that in principle, we do not accept any late submission. We encourage you to cooperate with each other in solving the problem sets. However, you should write down your own solutions by yourselves. You are welcome to flag confusing topics in the problem sets; this will not lower your grade. Some problems may require programming like CVX that runs in MATLAB. We will also provide a tutorial for programming when a first related-homework is issued. Exams As usual, there will be two exams: midterm and final. Please see syllabus for the schedule that our institution assigns by default - that is Thursday 9am-noon on an exam week for this course. Please let us know if someone cannot make it for the schedule. If your reason is reasonable, then we can change the schedule or can give a chance for you to take an exam in a different time slot that we will organize individually. Two things to notice. First, for both exams, you are allowed to use one cheating sheet, A4-sized and double-sided. So it is a kind of semi-closed-book exam. Second, for your convenience, we will provide an instruction note for each exam, which contains detailed guidelines as to how to prepare for the exam. Such information includes: (1) how many problems are in the exam; (2) what types of problems are dealt with in what contexts; (3) the best way to prepare for such problems. Course grade Here is a rule for the course grade that you are mostly interested in perhaps. The grade will be decided based on four factors: problem sets (22%); midterm (32%); final (40%); and interaction (6%). Here the interaction means any type of interaction. So it includes attendance, in-class participation, questions, discussion, and any type of interaction with me. Overview Now let's move onto the second part. Here is information for reading materials: Course Note (CN for short) 1 and Chapter 1 in LB. In this part, I will tell you how the theory of convex optimization was developed in which contexts. I will then provide you with specific topics that we will learn about through the course. Optimization To talk about a story of how the theory of convex optimization was developed, we need to first know about the history of optimization which includes convex optimization as a special case.
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