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IE 531 Linear Programming

Spring 2015

Sungsoo Park

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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)

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.

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Course Objectives

Why need to study LP?Important tool by itselfTheoretical basis for later developments (IP, Network, Graph, Nonlinear, schedul-

ing, Sets, Coding, Game, … )Formulation + package is not enough for advanced applications and interpretation

of results

Objectives of the class:Understand the theory of linear optimizationPreparation for more in-depth optimization theoryModeling capabilitiesIntroduction to use of software (Xpress-MP and/or CPLEX)

TopicsIntroduction and modelingSystem of linear inequalities, polyhedral theorySimplex method, implementationDuality theorySensitivity analysisDelayed column generation, Dantzig-Wolfe decomposition, Benders’ decomposi-

tionCore concepts of interior point methods

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Brief History of LP (or Optimization) Gauss: Gaussian elimination to solve systems of equations

Fourier(early 19C) and Motzkin(20C) : solving systems of linear in-equalities

Farkas, Minkowski, Weyl, Caratheodory, … (19-20C): Mathematical structures related to LP (polyhedron, systems of alterna-

tives, 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 op-timization, development of computer

1960s : more developments

1970s : disappointment, NP-completeness, more realistic expecta-tions

Khachian (1979) : ellipsoid method for LP

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1980s : personal computer, easy access to data, willingness to use mod-els

Karmarkar (1984) : Interior point method

1990s : improved theory and software, powerful computerssoftware 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 program-ming)

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, ...)

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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

…

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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 Program-ming, Networks, European Journal of Operational Research, Naval Research Logistics, Journal of the Operational Research Society, In-terfaces, …

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 vec-tors 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 .

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(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

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(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 (undi-rected)

: maximum value of the numbers and

: the element among which attains the value

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