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ECE 6397, Fall, 2012 Selected Topic in Optimization

ECE 6397, Fall, 2012 Selected Topic in Optimization. Zhu Han Department of Electrical and Computer Engineering Class 1 Aug. 27 nd , 2012. Outline. Instructor information Motivation to study optimization Course descriptions and textbooks What you will study from this course Objectives

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ECE 6397, Fall, 2012 Selected Topic in Optimization

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  1. ECE 6397, Fall, 2012Selected Topic in Optimization Zhu Han Department of Electrical and Computer Engineering Class 1 Aug. 27nd, 2012

  2. Outline • Instructor information • Motivation to study optimization • Course descriptions and textbooks • What you will study from this course • Objectives • Coverage and schedule • Homework, projects, and exams • Other policies • Reasons to be my students • Background and Preview

  3. Instructor Information • Office location: Engineering 2 W302 • Office hours: M 10am-2pm or by appointment • Email: zhan2@mail.uh.edu or hanzhu22@gmail.com • Phone: 713-743-4437(o), 301-996-2011(c) • Course website: • http://www2.egr.uh.edu/~zhan2/ECE6397/ • Research interests: • http://www2.egr.uh.edu/~zhan2 Wireless Networking, Signal Processing, and Security • http://wireless.egr.uh.edu/

  4. Motivations • Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints. • Interdisciplinary • Architecture • Nutrition • Electrical circuits • Economics • Transportation • Examples: • Determining which ingredients and in what quantities to add to a mixture being made so that it will meet specifications on its composition • Allocating available funds among various competing agencies • Deciding which route to take to go to a new location in the city

  5. Course Descriptions • What is the optimization framework? • What are the major types? • Convex vs. non-convex • Continuous vs. discrete • Centralized vs. distributed • Deterministic vs. stochatic • What are the theorems? • What are the applications? • What are the state-of-art research? • Can I find research topics? • How to conduct research and write technique paper

  6. Textbook and Software • Require textbook: • Zhu Han, DusitNiyato, WalidSaad, Tamer Basar, and Are Hjorungnes, Game Theory in Wireless and Communication Networks: Theory, Models and Applications, Cambridge University Press, UK, 2011. • Steven Boyd’s videos for convex optimization • Handout for parts of book, Zhu Han and K. J. Ray Liu, Resource Allocation for Wireless Networks: Basics, Techniques, and Applications, Cambridge University Press, 2008. • Other handouts • Require Software: MATLAB http://www.mathworks.com/ or type helpwin in Matlab environment

  7. Schedule • Introduction to optimization • Convex optimization • Steven Boyd’s class • http://www.stanford.edu/~boyd/cvxbook/ • 30% of the class • Need to watch videos as homework (17 videos for 1 hour 15 min each) • Watch the video before the class!!! • Class is just review • Integer/Combinatorial optimization • Might based on Georgia tech class • 15% • Stochastic optimization • Might based on UIUC class • 15% • Game Theory • based on my book • 40%

  8. Homework, Project, and Exam • Homework • Watch videos for convex optimization • Some other homework • Projects: simple MATLAB programs • Based on the simulation at the end of each chapter • Exams • Two independent exams • Grading policy • Participations • Attendance and Feedback • Quiz if the attendance is low

  9. Teaching Styles • Slides plus black board • Slides can convey more information in an organized way • Blackboard is better for equations and prevents you from not coming. • Course Website • Print handouts with 3 slides per page before you come • Homework assignment and solutions • Project descriptions and preliminary codes • Feedback • Too fast, too slow • Presentation, Writing, English, …

  10. Other Policies • Any violation of academic integrity will receive academic and • possibly disciplinary sanctions, including the possible awarding • of an XF grade which is recorded on the transcript and states that • failure of the course was due to an act of academic dishonesty. • All acts of academic dishonesty are recorded so repeat offenders • can be sanctioned accordingly. • CHEATING • COPYING ON A TEST • PLAGIARISM • ACTS OF AIDING OR ABETTING • UNAUTHORIZED POSSESSION • SUBMITTING PREVIOUS WORK • TAMPERING WITH WORK • GHOSTING or MISREPRESENTATION • ALTERING EXAMS • COMPUTER THEFT

  11. Reasons to be my students • Wireless Communication and Networking have great market • Usually highly paid and have potential to retire overnight • Highly interdisciplinary • Do not need to find research topics which are the most difficult part. • Research Assistant • Free trips to conferences in Alaska, Hawaii, Europe, Asia… • A kind of nice (at least looks like) • Work with hope and happiness • Graduate fast

  12. Different Kinds of Optimization

  13. Optimization Formulation and Analysis • We discuss how to formulate the problem as an optimization issue. • Specifically, we study what the objectives are, what the parameters are, what the practical constraints are, and what the optimized performances across the different layers are. • The tradeoffs between the different optimization goals and different users' interests are also investigated. • The goal is to provide the students a new perspective from the optimization point of view for variety of problems in engineering fields.

  14. Mathematical Programming • If the optimization problem is to find the best objective function within a constrained feasible region, such a formulation is sometimes called a mathematical program. • Many real-world and theoretical problems can be modeled in this general framework. • We discuss the four major subfields of the mathematical programming: • linear programming, • convex programming, http://www.stanford.edu/~boyd/cvxbook/ • nonlinear programming, • dynamic programming.

  15. What do we optimize? • A real function of n variables • with or without constrains • Without constraint • With constraint

  16. Lets Optimize • Suppose we want to find the minimum of the function • What is special about a local max or a local min of a function f (x)? at local max or local min f’(x)=0 f”(x) > 0 if local min f”(x) < 0 if local max

  17. Review max-min for R3

  18. Integer/Combinatorial Optimization • The discrete optimization is the problem in which the decision variables assume discrete values from a specified set. • The combinatorial optimization problems, on the other hand, are problems of choosing the best combination out of all possible combinations. • Most combinatorial problems can be formulated as integer programs. • Integer optimization is the process of finding one or more best (optimal) solutions in a well defined discrete problem space. • The major difficulty with these problems is that we do not have any optimality conditions to check if a given (feasible) solution is optimal or not. • We listed several possible solutions such as • relaxation and decomposition, • enumeration, • knapsack problem • cutting planes.

  19. Example of Integer Program(Production Planning-Furniture Manufacturer) • Technological data: Production of 1 table requires 5 ft pine, 2 ft oak, 3 hrs labor 1 chair requires 1 ft pine, 3 ft oak, 2 hrs labor 1 desk requires 9 ft pine, 4 ft oak, 5 hrs labor • Capacities for 1 week: 1500 ft pine, 1000 ft oak, 20 employees (each works 40 hrs). • Market data: • Goal: Find a production schedule for 1 week tomaximize the profit.

  20. Production Planning-Furniture Manufacturer: modeling the problem as integer program The goal can be achieved by making appropriate decisions. First define decision variables: Let xt be the number of tables to be produced; xc be the number of chairs to be produced; xd be the number of desks to be produced. (Always define decision variables properly!)

  21. Production Planning-Furniture Manufacturer: modeling the problem as integer program • Objective is to maximize profit: max 12xt + 5xc + 15xd • Functional Constraints capacity constraints: pine: 5xt + 1xc + 9xd  1500 oak: 2xt + 3xc + 4xd  1000 labor: 3xt + 2xc + 5xd  800 market demand constraints: tables: xt ≥ 40 chairs: xc ≥ 130 desks: xd ≥ 30 • Set Constraints xt , xc , xd  Z+

  22. Solutions to integer programs • A solution is an assignment of values to variables. • A feasible solution is an assignment of values to variables such that all the constraints are satisfied. • The objective function value of a solution is obtained by evaluating the objective function at the given point. • An optimal solution (assuming maximization) is one whose objective function value is greater than or equal to that of all other feasible solutions. • There are efficient algorithms for finding the optimal solutions of an integer program.

  23. Game Theory • Game theory is a branch of applied mathematics that uses models to study interactions with formalized incentive structures (“games"). • It studies the mathematical models of conflict and cooperation among intelligent and rational decision makers. • Rational means that each individual's decision-making behavior is consistent with the maximization of subjective expected utility. • Intelligent means that each individual understands everything about the structure of the situation, including the fact that others are intelligent rational decision makers. • We have discussed four different types of games, namely, the non-cooperative game, repeated game, cooperative game, and auction theory. • Slides http://wireless.egr.uh.edu/research.htm • The basic concepts are listed and simple examples are illustrated.

  24. Game Theory Overview • What is game theory? • The formal study of conflict or cooperation • Modeling mutual interaction among rational decision makers • Widely used in economics • Components of a “game” • Rational playerswith conflicting interests or mutual benefit • Strategies or actions • Utility as a payoff of player’s and other players’ actions • Outcome: Nash Equilibrium • Many types • Non-cooperative game theory • Cooperative game theory • Dynamic game theory • Stochastic game • Auction theory

  25. Rich Game Theoretical Approaches • Non-cooperative static game: play once • Mandayam and Goodman (2001) • Virginia tech • Repeated game: play multiple times • Threat of punishment by repeated game. MAD: Nobel prize 2005. • Tit-for-Tat (infocom 2003): • Dynamic game: (Basar’s book) • ODE for state • Optimization utility over time • HJB and dynamic programming • Evolutional game (Hossain and Dusit’s work) • Stochastic game (Altman’s work) Prisoner Dilemma Payoff: (user1, user2)

  26. Auction Theory Book of Myerson (Nobel Prize 2007), J. Huang, H. Zheng, X. Li

  27. Term Project • Forming the group, 2-3 people per group • Similar research background • Formulation of problems • Is that a problem? • What is the objective and constraints • What is best optimization techniques • Simulation • Matlab • Victim algorithm • Analysis • Writing • It will be a headache for everybody

  28. Homework • Convex optimization I • http://www.youtube.com/watch?v=McLq1hEq3UY • Watch before Wed. class!!! • Form Term project group • 2-3 people per group • Let me know in the next class for grouping and basic interests

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