Courses

• List of Courses
• Detailed Description

Graduate Studies

• Learning Goals
• Admission
• Masters Program
• Doctoral Program
• Financial Support
• Ph.D. Requirements
• Ph.D. Examination
• Research Proposal
• Student Research
• Thesis Defense
• Transfer of Credit
• Additional Information

Undergraduate Studies

• General Information
• Minor Requirements

 

Course Details

Integer Programming

Prerequisites: Calculus, Linear Algebra and Linear Programming

Course Outline:
Overview of discrete optimization models occurring in business, engineering, industry and the sciences Modelling with integer variables Specially structured problems:knapsack, covering and partitioning problems A quick introduction to complexity theory:problems, instances, worst-case complexity, polynomial algorithms, the classes P and NP Linear programming relaxations, integrality ofsolutions, unimodularity and applications for assignment problems, shortest path and network computations Enumerative methods:branch-and-bound, implicit enumeration, bounding techniques, Lagrangean and surrogate duality Cutting planes, Gomory's algorithm, lifting and projecting for binary optimization Heuristics: greedy algorithms, local search, truncated exponential schemes.

Book to be used: Integer Programming, Laurence A. Wolsey John Wiley & Sons, Inc. 1998 ISBN: 0-471-28366-5

* Required course for the Minor in Operations Research

Theory of Boolean Functions

Course Outline:
The course will describe the uses, the theory and the algorithmic aspects of Boolean and of pseudo-Boolean functions, using frequently occurring models from operations research, electrical engineering, and modern applied mathematics. Among the important problems to be studied, we mention the Satisfiability problem, dualization, determination of prime implicants of Boolean functions, logic minimization, quadratic 0-1 optimization, roof-duality, MAX-SAT, etc. Important classes of Boolean functions (monotonic, regular, Horn, threshold, etc.) and of pseudo-Boolean functions (super-modular, linearly separable) will be examined.
Numerous applications will be presented to computer engineering, discrete optimization, artificial intelligence, voting, game and reliability theory, etc. Strong connections between linear programming, graph theory, Boolean and pseudo-Boolean functions, in the development of algorithms for solving operations research problems will be emphasized.

Prerequisites: Familiarity with graph theory and linear programming is useful, but not absolutely necessary.

Computational Methods of Operations Research

Instructor: Endre Boros

Course Outline:
The course will be highly interactive with individual and group assignments and with intensive computer practice. The students will be offered various problems and projects to work on during the semester. Some of the projects will involve the use of certain software packages, while some others will require coding. In addition, each of the students will be required to solve homework assignments, mostly programming tasks. Grading will be based on homeworks and projects. The course concentrates on OR modeling and problem solving with AMPL, a mathematical modeling language. Additionally, elements of other programming environments will be described, and a few assignments will be given, in particular, in PERL to realize basic data structures and combinatorial algorithms; in C++ to develop basic routines, and interface with CPLEX and/or XPressMP; and on HTML, Javascript and CSS to develop home pages and interactive web-projects.

A sample of related books:
Robert Fourer, David M. Gay, and Brian W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, (Duxbury Press/Brooks/Cole Publishing Company, 1993)
MOSEL/XPressMP (Dash Optimization)
Daniel Gilly and the staff of O'Reilly & Associates, Inc., UNIX in a nutshell, (O'Reilly & Associates, Inc., Sebastopol, CA, 1992)
D.E. Knuth, The Art of Computer Programming: Fundamental Algorithms, (Addison Wesley, Reading, MA, 1973)
Matematika, User's Manual, (Wolfram Research, Inc., 1990)
Chuck Musciano and Bill Kennedy, HTML - The Definitive Guide, (O'Reilly & Associates, Inc., Sebastopol, CA, 1997)
H. Schildt, C -- The Pocket Reference, (Osborne McGraw-Hill, Berkeley, 1989)
Bjarne Stroustrup, C++ Programming Language, (Addison-Wesley, Reading, MA, 1997)
A. Thesen, Computer Methods in Operations Research (Adademic Press 1978)
Larry Wall, Tom Christiansen and Randal L. Schwartz, Programming Perl, (O'Reilly & Associates, Inc., Sebastopol, CA, 1996)

Stochastic Models in Operations Research

Instructor: Michael N. Katehakis
mnk@rutgers.edu (848-445-8181)
Place and Time: RUTCOR Bldg., Room 139 Friday, 12:00-3:00PM
Prerequisites: Probability Theory (640:477 or equivalent)

Topics:
1. Elements of probability theory: review of basics.
2. Definition of a stochastic process. Classification of stochastic processes.
3. Definition of a renewal process and a Poisson process.
4. Properties of the exponential random variable.
5. Poisson processes.
6. Renewal theory.
7. Discrete time Markov chains. classification of states. limit theorems. stationary distribution. applications.
8. Continuous time Markov chains. Kolmogorov equations. Birth and death processes. Stationary distributions. Applications.
9. Queueing systems. Basic notions. Markovian queues. M/G/1 type models. Applications.
10. Basic inventory models.
11. Martingales and applications to finance. (Time permitting).

Book: Sheldon M. Ross, “Applied Probability Models with Optimization Applications” Dover Publications, soft cover, 1992. ISBN: 0-486-67314-6

Grading: Based on the solutions of the homework problems and the results of the written midterm and final exams.
Office hours: After class or by appointment.

Reference Books:
• Ross, Introduction to Probability Models, Academic Press, 7th edition.
• Taylor and Karlin, An Introduction to Stochastic Modeling Academic press 1993 edition.
• Ross, Stochastic Processes, Wiley & Sons, 1982 or second edition 1996.
• Heyman and Sobel, Stochastic Models in Operations Research, McGraw-Hill, 1982, Vols. I, II.
• Barlow and Proschan, Statistical Theory of Reliability, HoIt, Rinehart and Winston, 1975.
• Prekopa, Special Topics in Stochastic O.R., Lecture Notes, RUTCOR.

Dynamic Programming

Instructor: Andrzej Ruszczyński
E-Mail: rusz@business.rutgers.edu
Web: http://www.rusz.rutgers.edu/

Topics:
The shortest path problem.The principle of optimality. Examples
Label correcting algorithms
Controlled Markov chains. Finite horizon stochastic problems
Dynamic programming equations. Applications
Discounted infinite horizon problems
Value and policy iteration methods. Linear programming approach
Applications in inventory control, scheduling, logistics
The multiarmed bandit problem
Undiscounted infinite horizon problems. Stochastic shortest paths
Methods for solving undiscounted problems
Optimal stopping; asset pricing
Average cost problems
Methods for solving average cost problems
Controlled continuous time Markov chains
Introduction to approximate dynamic programming

Textbooks
Main:
1. M.L. Puterman, Markov Decision Processes, John Wiley & Sons, 2005

Supplementary:
2. D.P. Bertsekas, Dynamic Programming and Optimal Control, Athena, 1995 or 2001 vol. I and II.
3. W. B. Powell, Approximate Dynamic Programming, John Wiley & Sons, 2007.

Grading
The course grade will be based on the following components:
1. Homework assignments (30%)
2. Computational projects (20%)
3. Midterm exam (20%)
4. Final exam (30%)

Discrete Optimization

Instructor: Dan Stratila

Prerequisites: 16:198:521 Linear Programming, Mathematical maturity (calculus, linear algebra)
(students who have not taken 16:198:521 may obtain permission from instructor)

Syllabus:

Background: Review of linear programming, the ellipsoid algorithm. Review of graphs and networks. Elements of complexity theory. Boolean and pseudo-Boolean functions, polyhedral theory and integer lattices.

Models: Knapsack, set-covering (vehicle routing, airline crew scheduling), packing, partitioning, clustering (stability number, voting districts, cutting-stock), bin-packing, scheduling (job-shop, flow-shop, one-machine), plant location, transportation problems.

General and 0-1 Integer Programming: Formulations, LP relaxation, rounding, unimodularity. Implicit enumeration and branch-and-bound tecniques. Polyhedral description: vertices, feasibility, valid inequalities, faces and facets. Value function, surrogate, subadditive and Lagrangean duality. Cutting planes, Gomory’s algorithm, Chvatal cuts, Lovasz-Schrijver cutting planes. Decomposition methods. Preprocessing techniques and heuristics.

Graphs& Networks: Minimum paths, spanning trees, max flow-min cut, CPM networks, matching algorithms. Traveling salesman, Chinese postman.

Matroids, Submodular Functions: Matroids, polymatroids, “greedy” algorithms, submodular functions, submodular function minimization.

References
• G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, J. Wiley, 1988.
• A. Schrijver, Theory of Linear and Integer Programming, J. Wiley, 1987.
• T. Lengauer, Combinatorial Optimization in Circuit Layout Design, J. Wiley, 1990.
• C.H. Papadimitriou and K. Stieglitz, Combinatorial Optimization: Algorithms and Complexity Prentice-Hall, 1982.
• L.A. Wolsey, Integer Programming, J. Wiley, 1998.
• W.F. Cook, W.H. Cunningham, W.R. Pulleyblank and A. Schrijver, Combinatorial Optimization, J.Wiley, 1998.

Stochastic Programming

Instructor: Andras Prekopa

Prerequisites: Probability theory, linear programming

Course Outline:
Overview of statistical decision principles.Overview of stochastic programming model constructions: reliability type models, penalty type models, mixed models, static and dynamic type models. The simple recourse model and its numerical solution techniques. Convexity theory of probabilistic constrained models.Bounding and approximation of probabilities. Numerical solution of probabilistic constrained models. Two-stage programming under uncertainty and the solution of the relevant problem by Benders' decomposition. Multi-stage stochastic programming models. Scenario aggregation. Distribution theory of stochastic programming. Applications to production, inventory control, water resources, finance, power and communication systems.
Stochastic programming is a rapidly developing area. It is on the border line of statistics and probability theory on the one hand and mathematical programming on the other hand. It can be defined as a methodology to optimize the design and operation of stochastic systems by the use of mathematical programming tools.

Book: A. Prekopa, Stochastic Programming, Academic Publishers 1995. (The instructor will make available a few printed copies for the audience.)

Grading: Based on the solutions of the homework problems and the quality of a term project work. Each student will have his/her own course-work subject following an agreement with the instructor.

Pseudo Boolean Functions

Instructors: Endre Boros

Course Outline:
Introduction, definitions and notations.
Examples:
General Pseudo-Boolean Functions - Representations of pseudo-Boolean functions, rounding procedures and derandomization, persistency, local optima, basic algorithm, reductions to quadratic optimization, posiform maximization, l2-approximations and applications to game theory.
Quadratic Pseudo-Boolean Functions - Roof duality (majoriztion, linearization, complementation, equivalence of formulations and persistency, network flow model), hierarchy of bounds (cones of positive quadratic pseudo-Boolean functions, complementation, majorization, linearization), polyhedra, heuristics.
Special classes - Sub- and supermodular functions, half-products, hyperbolic pseudo-Boolean functions, products of linear functions.
Approximation algorithms - MAX-SAT and variants, -approximation of MAX-2SAT via roof-duality, -approximation of MAXSAT via convex majorization.

Stochastic Models of Operations Research

Instructor: Myong K.(MK) Jeong (mjeong@rci.rutgers.edu; phone:732-445-4858; www.rci.rutgers.edu/~mjeong)
Office Hours:T, F after class, or by arrangement (Office: RUTCOR Room 115)

Prerequisites: Probability Theory (01:640:477 or 01:960:381)

Textbook: Sheldon M. Ross, An Introduction to Probability Models, 9th edition, Academic Press, 2007. ISBN: 0-12-598062-0.

Course Webpage: Sakai

Topics: Markov chains: definition, transition probabilities, special Markov chains (random walks, dams and inventories, branching processes), classification of states, limit theorems.
The Exponential distribution and the Poisson processes: derivations, homogeneous, non-homogeneous processes, spacial and marked Poisson processes.
Continuous time Markov chains: the Chapman-Kolmogorev equation, birth and death processes, the case of a finite state space, special cases, limiting behavior.
Renewal processes and their applications: definition, the renewal function, replacement models, renewal theorems, inspection paradox, applications.
Queueing theory: queueing systems, Little’s formula, Poisson arrivals and exponential and general service times, the case of an infinite number of servers, priority queues, queueing systems.
Brownian motions: definition, processes with independent increments, the maximum variable and the reflection principle, Brownian bridge, geometric Brownian motion, applications in modern financial theory.

Grading: Homework (25%), midterms (25%), and final exam (40%), and a team project with an oral presentation (10%)

Student Responsibility: Students are responsible for attending class, being on time, reading the material covered in lecture, and actively participating in class discussion.

Algorithmic Graph Theory and Its Applications

Instructor: Martin Charles Golumbic

Prerequisites: This is a graduate elective course. Students are expected to have mastered basic undergraduate knowledge of algorithms, complexity analysis and discrete mathematics.

Course Outline:
This course will be a self-contained continuation of the course (Part I) given last year. We present classical and more recent theory and algorithms in graph theory, in particular, related to intersection graphs and other related families of graphs. Intersection graphs are an extremely useful mathematical model for many applications in computer science, operations research, and even molecular biology. We begin by explaining and motivating the concept, and provide examples from the literature. The graph coloring problem on special families of these intersection graphs will be studied. Many of these techniques can be applied to scheduling classrooms or airplanes, allocating machines or personnel to jobs, or designing circuits. Rich mathematical problems also arise in the study of intersection graphs. We will present spectrum research results, from simple to sophisticated, which should interest both students and professors. Lectures will be chosen from the following list of topics.
Introduction to Intersection Graphs - [Go1, Mc21]
Review of Basic Graph Algorithms and Complexity Analysis - [Go2, CLR]
Chordal Graphs and their Applications - [Go4, Mc22]
Transitively Orderable Graphs - [Go5, Mc27.6]
Permutation Graphs - [Go7, Mc27.4]
Weakly Chordal Graphs and Strongly Chordal Graphs - [Mc27.3, Mc27.12, Go13]
Split Graphs, Threshold Graphs and CoGraphs - [Go6, Mc2 2.5, Go10 Mc25, Mc27.9]
Interval Graphs - [Go8, Mc23]
Tolerance Graphs - [GT1, GT2]
Interval Probe Graphs - [GT4, Mc23.4.1]
Intersection Graphs on Trees - [GT10, Mc22.3]

Texts:
Go: M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, second edition Elsevier (2004).
Chapters 1; 2; 4:1-4,6; 5:1,4,6-7; 6:1-2; 7:1-2,4-5; 8:1-4; 10*; 13
GT: M.C. Golumbic & A.N. Trenk, Tolerance Graphs, Cambridge University Press (2004).
Chapters 1; 2:1-3,6-7; 4:1-4,7-8; 10* (*optional)

Other references:
MP : Mahadev & Peled, Threshold Graphs and Related Topics
BLS: Brandstadt, Le & Spinrad, Graph Classes: A Survey
Mc2: T.A. McKee & F.R. McMorris, Topics in Intersection Graph Theory
Misc. papers from the literature

Students are responsible for reading all assigned material. Exercises are to be presented in class. Each student will also be expected to concentrate on a paper from the literature to be assigned in class and do a course project or presentation. The exam will be open book, covering the material discussed in the lectures.

Topics in Applied Operations Research

Instructor: Adi Ben-Israel, RUTCOR Bldg., Room 113

Prerequisites: Linear programming, probability, linear algebra, a year of calculus, and a course that used calculus or permission of the instructor.

Desirable background: Nonlinear programming. Some experience in simulation (for example,YASAI).

Course webpage:http://ben-israel.rutgers.edu/711/Course.htm
E-mail:adi.benisrael@gmail.com
Tel: 732-445-5631
Webpage: http://ben-israel.rutgers.edu/index.htm

Course Outline: The objectives of the course are (a) to introduce selected models and applications of Operations Research (O.R.), (b) establish (review, and if necessary cover) the theory needed for these applications, and (c) study the human issues (political, cultural, sociological etc.) relevant for implementing and validating a policy.
O.R. is used here in the wide sense, borrowing as needed from Optimization, Probability, Statistics and Computing. Applications are selected because they are representative, important, and/or not covered by other courses offered by our program. Possible topics and applications:
1. Paradoxes and weird phenomena in probability, statistics and O.R.
2. Location problems, single and multiple facilities.
3. Search engines. The page rank algorithm.
4. Models of risk and uncertainty.
5. Forecasting. Political elections.
6. Trends. Demographics.
7. Systematic climate changes. Policy implications.
8. Nonrenewable natural resources. Optimal rate of extraction. Oil. Coal.
9. Renewable natural resources. Forestry. Fisheries. Agriculture. Solar and wind energy.
10. Health care. Insurance.
11. Epidemics and vaccination models.
12. Dynamic pricing.
13. Traffic systems. Flow. Congestion.
14. Clustering and classification.
15. Bayesian analysis.
16. Statistics and law.
17. Maintenance and replacement policies.
These topics will be studied and illustrated using appropriate case studies and histories (e.g., FEDEX for topic 2, GOOGLE for topic 3, etc.), and selected readings from the research literature.
Students will choose topics, search the literature, prepare lectures, and present them in a seminar setting starting from the 3rd week of the course.
There will be projects, where groups of students work on actual or hypothetical applications and present their reports. The projects must be finished and presented by the last day of classes.
The grade in the course will be based on the student's participation in class, the seminar presentation, and the project.

Introductory Topics in OR

Course Outline: This course is an introduction to the basic methods and models of Operations Research (abbreviated O.R.): linear programming, network models, integer programming, nonlinear programming, inventory control, dynamic programming.
O.R. can be described as follows (see text, p. 1) : "... the term operations research (or, often, management science) means a scientific approach to decision making, which seeks to determine how to design and operate a system, usually under conditions requiring the allocation of scarce resources."
The course is computer based, using MS Excel, and Solver (an add-in to Excel).
The study of the methodology of O.R. requires a working knowledge of linear algebra and probability, and a good understanding of calculus. These are expected of the students.

Text: Operations Research, Applications and Algorithms (3rd edition), Wayne L. Winston, Duxbury Press

Nonlinear Programming

Instructor: Andrzej Ruszczynski

E-Mail: rusz@business.rutgers.edu

Web: http://www.rusz.rutgers.edu/

Course Outline:
1. Convex sets. Separation.
2. Cones.
3. Convex functions.
4. Elements of subdifferential calculus.
5. Tangent cones. Metric regularity.
6. Optimality conditions.
7. Lagrangian duality.
8. The method of steepest descent.
9. Newton's method.
10. Conjugate gradient methods.
11. Nongradient methods. Truncated Newton's method.
12. Feasible direction methods.
13. Penalty methods.
14. Dual and augmented Lagrangian methods.
15. Sequential quadratic programming.
16. Interior point methods.
17. Introduction to Nondifferentiable Optimization.

Text: A. Ruszczynski, Nonlinear Optimization, Princeton University Press, Princeton 2006 (ISBN-10: 0691119155, ISBN-13: 978-0691119151)

Grading: The final grade will be based on homework assignments, involving theoretical problems and computational projects, and on the take-out final exam.

Queueing Theory

Prerequisites: Probability and elementary knowledge of Stochastic Processes or Stochastic Models.

Course Outline:
Elements of Stochastic Processes: Definition of a stochastic process; strict and covariance stationarity; ergodicity; discrete time Markov chains; semiMarkov processes and chains; embedded Markov chains; continuous time Markov chains; birth and death processes; Poisson processes; diffusion processes; Markovian queueing processes and Jackson type networks.
Definition of a queueing system; Little's theorem; DAASSP (departures and arrivals see same picture) theorem; the random observer; ROSTA (random observer sees time averages), PASTA and ASTA theorems; the work in the system; the random modification (remaining work in service); work conserving schemes (disciplines) and the work conservation theorem.
Applications: the relation between queueing time and queue size in M/G/1 type queues; busy period characteristics in conservative M/G/1 type queues; nonpreemptive priorities; optimal priority schemes; SRPF schemes.
Generalized M/G/1 queues:The generic model; M/G/1; a sequence of two servers in tandem with no intermediate queue; server's breakdowns and vacations; preemptive repeat and resume priorities; Takacs integro differential equation approach. A busy period approach; Gated queues with limited batch sizes, unlimited batch sizes, processor sharing, FIFO, LIFO and random selection for service.
Lindley's GI/G/1 model: the Wiener-Hopf type integral equation and its solution; bounds on the expected equilibrium waiting time; Kingmans's heavy traffic approximation; GI/M/1 example; Kendal's and Smith's theorems for the conditional equilibrium distributions of waiting times and lines in GI/M/r queues.
Comparison methods for queues: Stochastic ordering; convex and concave ordering; classes of distribution functions (IFR/DFR, etc.); montonicity and comparability of stochastic processes; monotonicity properties and bounds and approximations for queueing processes.
Tandem queues: Avi-Itzhak's theorem for deterministic service times; bounding and approximations; optimal servers ordering; just in time systems and throughput approximations; strategies for deployment of flexible servers; schemes for fork join (split and match) systems; correlated service times.
Networks:Reversibility, quasi reversibility, symmetric queues and product form (kelly type) networks; nonproduct form networks; approximations and bounds; decomposition and aggregation methods, numerical methods.
Applications of queueing models:Service systems; traffic and transportation; computer systems; voice and data communications; production and logistics. * It is not possible to cover all the listed topics in depth in a one semester course. Some of the topics will only be surveyed.

Grading: Home assignments, class presentations and midterm(s) (take home) 50 points, Final exam (possibly take home) 50 points

Reading: Home assignments include reading of a number of journal publications in addition to specific sections in the reference books.

Reference Books:
L. Kleinrock, Queueing Systems, Vol. 1: Theory, Wiley & Sons, 1975. (* Major reference book.)
L. Kleinrock, Queueing Systems, Vol. 2: Computer Applications, Wiley & Sons, 1976.
J.W. Cohen, The Single Server Queue, North Holland, 1982.
F.P. Kelly, Reversibility and Stochastic Networks, Wiley & Sons, 1979.
D.R. Cox and W.L. Smith, Queues, Wiley & Sons, 1961.
L. Takacs, Introduction to the Theory of Queues, Oxford University Press, 1962.
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, Wiley & Sons, 1985.
J. Walrand, An Introduction to Queueing Networks, Prentice Hall, 1988.
A.E. Conway and N.D. Georganas, Queueing Networks Exact Computational Algorithms, MIT Press, 1989.
R.W. Conway, W. L. Maxwell and L.W. Miller, Theory of Scheduling, Addison Wesley, 1967.
D. Gross and C.M. Harris, Fundamentals of Queueing Theory, Wiley & Sons, 1974.
D.P. Heyman and M.J. Sobel, Stochastic Models in Operations Research (two volumes). McGraw Hill, 1982.
J. Riordan, Stochastic Service Systems, Wiley & Sons, 1962.
R.W. Wolf, Stochastic Modeling and the Theory of Queues, Prentice Hall, 1989.

Game Theory

Instructor: Vladimir Gurvich

Prerequisites: None. Graph theory and linear programming would be useful but not necessary.

Email: gurvich@rutcor.rutgers.edu

Course Outline:
1. Matrix games, max min , min max and saddle point. Pure and mixed strategies. Solvability in mixed strategies. Von Neumann's Theorem for matrix games.
2. Bimatrix and n-matrix games. Nash equilibria and Nash solvability. Perfect equilibria and perfect solvability. Sophisticated equilibria and dominance solvability.
3. Games in extensive, positional and normal form. Perfect information and solvability in pure strategies. Nash solvability of the cyclic games.
4. Domination and dominance solvability. Backward induction. Dominance solvable extensive and secret veto voting schemes.
5. Cooperative games. Coalitions. Transferable and non-transferable utilities, TU- and NTU-games. Cores and core-solvability. Bondareva-Shapley's Theorem and Scarf's Theorem.
6. Effectivity functions and game forms, Moulin-Peleg's Theorem. Cooperative games in effectivity function form, Keiding's Theorem. Stable effectivity functions and stable families of coalitions.
7. Intrduction to Social Choice Theory. Paradox Arrow. Social choice functions and correspondences.
8. Boolean functions and graphs in game theory: Boolean duality and Nash solvability. Read-once Boolean functions, P4-free graphs and normal form of the positional games with perfect information. Stable effectivity functions and Berge's perfect graphs. Stable families of coalitions and normal hypergraphs. The Shapley value and the Banzhaf power index for cooperative games and approximation of pseudo-Boolean functions.

Simulation

Course Outline:
1. Introduction to discrete event simulation 1, 2
2. Random number generation, Intro. To ARENA 4, 5.1-5.3
3. ARENA basics 5.4--5.8
4. Model testing and debugging 6
5. Input analysis 7
6. Model verification and validation 8
7. Output analysis 9
8. Advanced ARENA modeling-production models 1 11
9. Advanced ARENA modeling-production models 2 11
10. Advanced ARENA modeling-transportation models 1 12
11. Advanced ARENA modeling-transportation models 2 12
12. Advanced ARENA modeling-information systems 1 13
13. Advanced ARENA modeling-information systems 2 13
14. Advanced ARENA modeling-other topics

Text: Simulation with Arena (2nd edition), W. David Kelton, Randall P. Sadowski, Deborah A. Sadowski, McGraw-Hill 2002

Semidefinite and Second Order cone Programming

Prerequisites: Ph.D. standing and the proverbial "mathematical maturity" are the only prerequisites. Knowledge of linear programming will be quite helpful though strictly speaking not required. Good understanding of linear algebra is essential.

Course Outline:
Semidefinite programming (SDP) is a relatively new topic in optimization theory that emerged in early nineties. It has attracted considerable attention for several reasons. First it is a natural model that pops up in diverse application areas from combinatorial optimization and graph theory, to statistics, to many areas of engineering. Second the underlying problem is now fairly well-understood and it is possible to come up with both theoretically and pragmatically efficient algorithms. Third, the subject has intellectual appeal in that methods from several areas of mathematics come together to form an elegant theory. In this course we present a rather comprehensive survey of the subject. We will overview semidefinite programming, study duality and complementarity conditions, and cover at least one class of interior point algorithms. We will survey several areas of applications. For instance in combinatorial optimization we study Lovasz theta function and Goemans-Williamson approximation algorithm to the MAXCUT and related problems. We will study one or two applications in statistics and finance. For instance we study the connection to Markowitz portfolio selection theory and to statistical optimal experiment design. We will also study the connection to positive polynomials and moment problems and theory implications in shape constrained approximation and regression. Our coverage will be mathematically deep, though almost all preliminary material will be reviewed. In particular we shall cover the theory of Euclidean Jordan algebras and their relevance to the most elegant formulation of SDP and its generalization.

Student Requirements: Each student is required to take turn and jot down notes during the lectures and then transcribe them into LaTeX. The notes will then be posted in the course web page. In addition each student is required to prepare a 30-40 minute talk which can be either presentation of a research paper by others or it can be a project that he/she has conducted in the course.

Reading: There is no official textbook. Relevant papers will be made available in the course home page.

Convex Analysis and Optimization

Instructor: Jonathan Eckstein

Office hours: Tuesdays 2:00 - 3:30 PM and by appointment.

Prerequisites: "Basic knowledge of finite-dimensional real analysis: open sets, closed sets, and convergence of sequences."

Course Outline:
Fundamentals of finite-dimensional convex sets and functions. Subgradient mappings, optimality conditions, and Lagrange multipliers. Set-valued monotone operators, conjugate functions, and optimization duality theory. Convex-analytic optimization algorithms such as subgradient, proximal, and bundle methods, including applications to nonconvex/discrete optimization.

Book: "Convex Analysis and Optimization", by Bertsekas, Nedic, and Ozdaglar, details at http://www.athenasc.com/convexity.html

Grading: Grades will be based on two take-home exams and homework assignments given out every one or two weeks.

Financial Mathematics

Instructor: Andras Prekopa

Prerequisites: Probability Theory, Linear Programming.

Course Outline: Cash flow streams. Financial instruments (stocks, bonds, futures, options, cash flows). Utility functions. Arbitrage pricing theory. Application of Martingales. Brownian motions. Ito's lemma. Black-Scholes theory. Parabolic PDEs and their numerical solutions. The Feynman-Kac solution. Exotic and path-dependent options (chooser, barrier, lookback, Asian, Bermudan, etc.). Interest rate models (Vasicek, Hull-White). Short introduction to stochastic programming models.Markowitz?s mean-variance models. Bond portfolio composition models. Term structures. The use of goal programming. Dynamic option selection models. Value at Risk models.

Books: Below is a list of books which are useful to study the subject. However, only parts of them will be used in the course. A complete manuscript of the presentations will be available.
1. D. Duffie, Dynamic Asset Pricing Theory, Second Edition, Princeton University Press, 1996.
2. A. Prekopa, Stochastic Programming, Kluwer, 1995.
3. P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives. Cambridge University Press, 1997.

Grading: Based on the homework problem solutions and the quality of a term project prepared by each student.

Theory of Linear Optimization

Instructor: Mariya Naumova

Prerequisite: 01:640:250 Introductory Linear Algebra

Topics include convex sets, polyhedra, Farkas lemma, canonical forms, simplex algorithm, duality theory, revised simplex method, primal-dual methods, complementary slackness theorem, maximal flows, transportation problems, 2-person game theory.

Students will have the chance to apply the methods to real life problems. One of the aims of the course will be to teach the students the path: from real life problem to abstraction, to mathematical formulation, to solving the mathematical problem, to applying this solution in the real life framework.

Book to be used: Linear Programming, Vasek Chvatal, W. H. Freeman and Company 1980. Available at the Bookstore on College Avenue. ISBN 0-7167-1587-2

Data Mining and Modeling and Its Applications in Engineering and Operations Research

Course 16:540:694 (CRN: 76385)
Instructor: Dr. Myong K. (MK) Jeong (mjeong@rci.rutgers.edu;www.rci.rutgers.edu/~mjeong)
Class time and place: Fri 1:40 –4:40 pm, SEC 207

Prerequisites: Linear programming, introductory statistics
Textbooks: No textbook is required and class materials will be provided by the instructor.
Grading: Homeworks and class project

Topics (The topics and level of their coverage will be determined based on the background and interests of the audience)

Data Mining Overview: Concepts, overview of main data mining tools, applications of data mining

Classification: Bayesian decision theory, Bayesian classifier; Linear (quadratic, flexible, regularized) discriminant analysis; Logistic regression model; Regularization (ridge regression, Lasso, and others); Mathematical programming for classification

Regression: Loss functions; Regularization (penalization), bias, variance, and model complexity; Ridge regression, Lasso, and others); Variable selection in regression, AIC, BIC, ICOMP; Robust regression

Clustering: Similarity measures; K-means, K-medois, Kernel K-means; Spherical K-means; Gaussian mixture models; Probabilistic clustering; Assessment of clustering results (validity index, internal/external criteria); Fuzzy clustering; Mathematical programming for clustering

Anomaly Detection (or Novelty Detection): One-class classification problem; Support vector data description; PCA data description

Feature Selection: Metrics and distance functions depending on data types (numeric, text, …); dynamic time warping (DTW); Distance between probability distributions (e.g., divergence); separability measures; Optimization methods for feature selection: branch and bound method, floating search, genetic algorithms; Other research issues: SVM-based feature selection, huge-scale feature selection

Kernel-based Methods for Classification, Regression, and Anomaly Detection: Support vector machines for classification; Kernel trick; Mercer’s theorems; Convex optimization; Gaussian processes; Relevance vector machine; Extension of kernel-based approaches to PCA, PLS, CCA, and others

Logical Analysis of Data (LAD) for Classification: Introduction of LAD and optimization models for LAD; Current research issues in LAD; Logical Analysis of Data (LAD) for Regression

Tree-based Methods: Classification and regression trees (CARTs); Variants of CARTs; Comparison between LAD and CARTs

Function approximation: Basis expansions; Splines; Wavelets

Transformation for Feature Extraction, Dimensionality Reduction: Principal component analysis (PCA), probabilistic PCA, principal curves and surfaces; Partial least squares (PLS), independent component analysis (ICA); Latent variables and factor analysis; Multidimensional scaling; Kernel-PCA, Kernel-PLS

Chemometrics: Spectra (or time series) data analysis; Preprocessing of data and dimensionality reduction; Feature extraction and selection with spectra data

Ensemble Methods: General idea; Bagging; Boosting (AdaBoost)

Image Data Mining: Introduction of hyperspectra image data, band selection and other research issues; fMRI data analysis and applications

Spatial Data Mining: Join counts; Spatial correlogram; Kriging; Markov random field

Other topics: Reinforcement Learning; Graphical models (Bayseian networks); Association rule, Recommendation systems; Sampling methods (MCMC, Importance sampling); Hidden Makov models; Neural networks;

Resource Allocation: Models, Algorithms, and Applications

Instructor: Hanan Luss
16:711:611 Index #64485 3 Credits
Time: Wed. 1:40-4:40PM Location: Room 139 or 166, RUTCOR Building

Prerequisites: Linear Programming, basic knowledge of nonlinear and integer programming. Course Outline: This graduate course (for PhD and qualified MS students) examines diverse resource allocation problems, where a limited amount of resources are allocated among competing activities. We focus on models with significant special mathematical structures, where the corresponding formulations can be exploited and solved by elegant, computationally efficient algorithms. Most of the course will emphasize diverse equitable resource allocation models, where numerous resources are allocated fairly among competing activities.

Sample of topics that will be covered include: (i) Framework of resource allocation models and, in particular, of equitable resource allocation. (ii) Allocation of a single resource among competing activities while maximizing the sum of concave return functions; (iii) Equitable allocation of multiple resources among competing activities (lexicographic minimax/maximin optimization); (iv) Extensions to allocation of substitutable resources; (v) Extensions to multi-period resource allocation; (vi) Equitable allocation of resources in multi-commodity network flows; (vii) Equitable content distribution in networks (e.g., video-on-demand); and (viii) Equitable resource allocation with discrete decision variables. We will explore the use of these models and algorithms in the context of various application areas, including logistics, communication networks, and facility location.

References: H. Luss, Equitable Resource Allocation: Models, Algorithms, and Applications, under preparation 2012 (book draft). T. Ibaraki and N. Katoh, Resource Allocation Problems: Algorithmic Approaches, The MIT Press, 1988. N. Katoh and T. Ibaraki, Resource Allocation Problems, in D-Z. Du and P. M. Pardalos (Editors), Handbook of Combinatorial Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998 (survey). H. Luss, On Equitable Resource Allocation Problems: A Lexicographic Minimax approach, Operations Research 47, 361-378, 1999 (survey). M. Patriksson, A Survey on the Continuous Nonlinear Resource Allocation Problem, European Journal of Operational Research 185, 1-46, 2008 (survey). A variety of published papers.

Stochastic Networks

Instructor: Michael Tortorella(mtortore@rci.rutgers.edu) Room 148, RUTCOR Bldg.

Office Hours: Thursdays 2-3 PM or by arrangement.

Synopsis: Networks underlie almost every facet of modern life. Telecommunications, electricity, oil and gas and water distribution, and logistics networks are but the most obvious examples. Network theory has also been fruitfully applied in biology, sociology, and finance. When demands on the network are random and/or elements of the network are unreliable, network design and performance analysis techniques need to be expanded to incorporate these essential features. This course provides an introduction to network design and performance analysis problems and solutions in stochastic networks.

Topics: Foundations, review of graph theory and stochastic processes. Survey of random graph theory. Network design and optimization. Flows in networks with unreliable elements. Queueing network models. Heavy traffic limit theorems. Incorporating unreliable network elements. Cross-network delay models. Path set and cut set models for capacitated stochastic networks. Network resiliency. Service reliability.

Prerequisites: Working knowledge of graph theory and stochastic processes

Books: R. F. Serfozo , Stochastic Networks. Springer 1999. (tentative) L. Kleinrock, Communication Nets. Dover 2007.

Grading: Based on a presentation made by the student on a paper or book chapter relevant to the course. Each student will choose his/her own material with consent of the instructor.