1 IE 531 Linear Programming - PowerPoint Presentation

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IE 531 Linear Programming

Spring 2018

Sungsoo

Park

Linear Programming 2018

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Instructor

Sungsoo

Park (room 4112,

sspark@kaist.ac.kr

,

tel:3121

)

Office hour: Mon, Wed 14:30 – 16:30 or by appointment

Classroom: E2-2 room 1120

Class hour:

Tu

,

Thr

14:30 – 16:00

Homepage:

http://solab.kaist.ac.kr

TA:

Kiho

Seo

(

emffp1410@kaist.ac.kr

)

,

Room: 4113, Tel: 3161

Office hour:

Tu

,

Thr

14:00

–

16:00

or by appointment

Grading: Midterm 30-40%, Final 40-60%, (closed book/notes)

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 (Chapter 1 on homepage)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 2018

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Course ObjectivesWhy 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 optimizationPreparation for more in-depth optimization theoryModeling capabilitiesIntroduction to use of software (Xpress-MP and/or CPLEX)

Topics

Introduction and modeling (1 week)

System of linear inequalities, polyhedral theory (4 weeks)Geometry of LP (1 week)Simplex method, implementation (2 weeks)

Midterm exam (1 week)

Duality theory (2 weeks)

Sensitivity analysis (1 week)

Delayed column generation, Dantzig-Wolfe decomposition, Benders decomposition (2 weeks)

Core concepts of interior point methods

(1 week)

Final exam (1 week)

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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 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

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1980s : personal computer, easy access to data, willingness to use models

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 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, ...)

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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

https://

www.informs.org/About-INFORMS/History-and-Traditions/OR-Application-Areas

…

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Resources

Societies:

INFORMS (the Institute for Operations Research and Management Sciences) :

https://www.informs.org

EURO : https://www.euro-online.org/web/pages/1/home MOS (Mathematical Optimization Society) : http://www.mathopt.org/

Korean Institute of Industrial Engineers :

http://kiie.org

Korean Operations

Research/Management Science

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, IIE Transactions,

Interfaces

, …

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

.

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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 subset of

)

:

is strictly contained in

(

is

proper subset

of

)

: 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 (undirected

)

: maximum value of the numbers

and

: the element among

which attains the value

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