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IE 531 Linear Programming

IE 531 Linear Programming. Spring 2014. Sungsoo Park. Instructor Sungsoo Park (room 4112, sspark@kaist.ac.kr , tel:3121 ) Office hour: Tue, Thr 14:30 – 16:00 or by appointment Classroom: E2 room 1120 Class hour: Tue, Thr 16:00 – 17:30 Homepage: http://solab.kaist.ac.kr

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IE 531 Linear Programming

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  1. IE 531 Linear Programming Spring 2014 Sungsoo Park

  2. Instructor • Sungsoo Park (room 4112, sspark@kaist.ac.kr, tel:3121) • Office hour: Tue, Thr 14:30 – 16:00 or by appointment • Classroom: E2 room 1120 • Class hour: Tue, Thr 16:00 – 17:30 • Homepage: http://solab.kaist.ac.kr • TA: • KihoSeo, JunghwanKwak(room 4113, emffp1410@kaist.ac.kr, junghoan@kaist.ac.kr, tel:3161) • Office hour: Mon, Wed 14:30 – 16:00 or by appointment • Grading: Midterm 30-40%, Final 40-60%, HW 10-20% (including Software CPLEX/Xpress-MP)

  3. Text: "Introduction to Linear Optimization" by D. Bertsimas and J. Tsitsiklis, 1997, Athena Scientific(not in bookstore, reserved in library) and class Handouts • Prerequisite: basic linear algebra/calculus, earlier exposure to LP/OR helpful, mathematical maturity (reading proofs, logical thinking) • No copying of the homework. Be steady in studying.

  4. Course Objectives • Why need to study LP? • Important tool by itself • Theoretical basis for later developments (IP, Network, Graph, Nonlinear, scheduling, Sets, Coding, Game, … ) • Formulation + package is not enough for advanced applications and interpretation of results • Objectives of the class: • Understand the theory of linear optimization • Preparation for more in-depth optimization theory • Modeling capabilities • Introduction to use of software (Xpress-MP and/or CPLEX)

  5. Topics • Introduction and modeling • System of linear inequalities, polyhedral theory • Simplex method, implementation • Duality theory • Sensitivity analysis • Delayed column generation, Dantzig-Wolfe decomposition, Benders’ decomposition • Core concepts of interior point methods

  6. Brief History of LP (or Optimization) • Gauss: Gaussian elimination to solve systems of equations Fourier(early 19C) and Motzkin(20C) : solving systems of linear inequalities Farkas, Minkowski, Weyl, Caratheodory, … (19-20C): Mathematical structures related to LP (polyhedron, systems of alternatives, polarity) Kantorovich (1930s) : efficient allocation of resources (Nobel prize in 1975 with Koopmans) Dantzig (1947) : Simplex method 1950s : emergence of Network theory, Integer and combinatorial optimization, development of computer 1960s : more developments 1970s : disappointment, NP-completeness, more realistic expectations Khachian (1979) : ellipsoid method for LP

  7. 1980s : personal computer, easy access to data, willingness to use models Karmarkar (1984) : Interior point method 1990s : improved theory and software, powerful computers software add-ins to spreadsheets, modeling languages, large scale optimization, more intermixing of O.R. and A.I. Markowitz (1990) : Nobel prize for portfolio selection (quadratic programming) Nash (1994) : Nobel prize for game theory 21C (?) : Lots of opportunities more accurate and timely data available more theoretical developments better software and computer need for more automated decision making for complex systems need for coordination for efficient use of resources (e.g. supply chain management, APS, traditional engineering problems, bio, finance, ...)

  8. Application Areas of Optimization • Operations Managements Production Planning Scheduling (production, personnel, ..) Transportation Planning, Logistics Energy Military Finance Marketing E-business Telecommunications Games Engineering Optimization (mechanical, electrical, bioinformatics, ... ) System Design …

  9. Resources • Societies: • INFORMS (the Institute for Operations Research and Management Sciences) : http://www.informs.org • MOS (Mathematical Optimization Society) : http://www.mathopt.org/ • Korean Institute of Industrial Engineers : http://kiie.org • Korean Operations Research Society : http://www.korms.or.kr • Journals: Operations Research, Management Science, Mathematical Programming, Networks, European Journal of Operational Research, Naval Research Logistics, Journal of the Operational Research Society, Interfaces, …

  10. Notation • : the set of real numbers : the set of vectors with real components : the subset of of vectors whose components are all : the set of integers : the set of nonnegative integers : the vector of with components . All vectors are assumed to be column vectors unless otherwise specified. , or : the inner product of and , . : Euclidean norm of the vector , . : every component of the vector is larger than or equal to the corresponding component of . : every component of the vector is larger than the corresponding component of .

  11. (continued) , or : transpose of matrix rank(): rank of matrix : the empty set (without any element) : the set consisting of three elements and : the set of elements such that … : is an element of the set : is not an element of the set : is contained in (and possibly ) : is strictly contained in : the number of elements in the set , the cardinality of : the union of the sets and : the intersection of the sets and , or : the set of the elements of which do not belong to

  12. (continued) such that: there exists an element such that such that: there does not exist an element such that : for any element of … (P) (Q): the property (P) implies the property (Q). If (P) holds, then (Q) holds. (P) is sufficient condition for (Q). (Q) is necessary condition for (P). (P) (Q): the property (P) holds if and only if the property (Q) holds , or : graph which consists of the set of nodes and the set of arcs (directed) , or : graph which consists of the set of nodes and the set of edges (undirected) : maximum value of the numbers and : the element among which attains the value

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