IE 531 Linear Programming Spring 2015 Sungsoo Park

Homepage: http://solab.kaist.ac.kr TA: Instructor Sungsoo Park (room 4112, sspark@kaist.ac.kr, tel:3121) Office hour: Tue, Thr 13:30 – 15:30 or by appointment Classroom: E2 room 1120 Class hour: Mon, Wed 14:30 – 16:00 Homepage: http://solab.kaist.ac.kr TA: Junghwan Kwak (junghoan@kaist.ac.kr), Seulgi Jung (sgjung@kaist.ac.kr) Room: 4113, Tel: 3161 Office hour: Mon, Wed 13:00 – 14:30 or by appointment Grading: Midterm 30-40%, Final 40-60%, HW 10-20% (including Software CPLEX/Xpress-MP) Linear Programming 2015

Prerequisite: basic linear algebra/calculus, 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. Linear Programming 2015

Course Objectives Why need to study LP? Objectives of the class: 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) Linear Programming 2015

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 Linear Programming 2015

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 Linear Programming 2015

Karmarkar (1984) : Interior point method 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), Roth, Shapley (2012) : 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, ...) Linear Programming 2015

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 … Linear Programming 2015

Resources Societies: Journals: 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, … Linear Programming 2015

Notation 𝑅: the set of real numbers 𝑅 𝑛 : the set of vectors with 𝑛 real components 𝑅 + 𝑛 : the subset of 𝑅 𝑛 of vectors whose components are all ≥0 𝑍: the set of integers 𝑍 + : the set of nonnegative integers 𝑥= 𝑥 1 ⋮ 𝑥 𝑛 : the vector of 𝑅 𝑛 with components 𝑥 1 ,…, 𝑥 𝑛 . All vectors are assumed to be column vectors unless otherwise specified. 𝑥 ′ 𝑦, or 𝑥 𝑇 𝑦: the inner product of 𝑥 and 𝑦, 𝑖=1 𝑛 𝑥 𝑖 𝑦 𝑖 . 𝑥 : Euclidean norm of the vector 𝑥, 𝑥 ′ 𝑥 . 𝑥≥𝑦: every component of the vector 𝑥 is larger than or equal to the corresponding component of 𝑦. 𝑥>𝑦: every component of the vector 𝑥 is larger than the corresponding component of 𝑦. Linear Programming 2015

𝐴′, or 𝐴 𝑇 : transpose of matrix 𝐴 rank(𝐴): rank of matrix 𝐴 (continued) 𝐴′, or 𝐴 𝑇 : transpose of matrix 𝐴 rank(𝐴): rank of matrix 𝐴 ∅: the empty set (without any element) 𝑥,𝑦,𝑧 : the set consisting of three elements 𝑥, 𝑦 and 𝑧 𝑥:𝑥 such that … : the set of elements 𝑥 such that … 𝑥∈𝑋: 𝑥 is an element of the set 𝑋 𝑥∉𝑋: 𝑥 is not an element of the set 𝑋 𝐴⊆𝑋: 𝐴 is contained in 𝑋 (and possibly 𝐴=𝑋) 𝐴⊂𝑋: 𝐴 is strictly contained in 𝑋 𝑋 : the number of elements in the set 𝑋, the cardinality of 𝑋 𝐴∪𝐵: the union of the sets 𝐴 and 𝐵 𝐴∩𝐵: the intersection of the sets 𝐴 and 𝐵 𝑋∖𝐴, or 𝑋−𝐴: the set of the elements of 𝑋 which do not belong to 𝐴 Linear Programming 2015

∃ 𝑥 such that: there exists an element 𝑥 such that (continued) ∃ 𝑥 such that: there exists an element 𝑥 such that ∄ 𝑥 such that: there does not exist an element 𝑥 such that ∀ 𝑥∈𝑋 … : for any element 𝑥 of 𝑋 … (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 𝐺=(𝑉,𝐴), or 𝑁=(𝑉,𝐴): graph 𝐺 (or 𝑁) which consists of the set of nodes 𝑉 and the set of arcs (directed) 𝐴 𝐺=(𝑉,𝐸), or 𝑁=(𝑉,𝐸): graph 𝐺 (or 𝑁) which consists of the set of nodes 𝑉 and the set of edges (undirected) 𝐸 𝑚𝑎𝑥⁡{𝑎,𝑏,𝑐} : maximum value of the numbers 𝑎,𝑏, and 𝑐 𝑎𝑟𝑔𝑚𝑎𝑥{𝑎,𝑏,𝑐}: the element among 𝑎,𝑏,𝑐 which attains the value 𝑚𝑎𝑥⁡{𝑎,𝑏,𝑐} Linear Programming 2015