1. IE 531 Linear Programming
Spring 2015
Sungsoo Park

2. Homepage: http://solab.kaist.ac.kr TA:
Instructor
Sungsoo Park (room 4112, 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:
TA:
Junghwan Kwak Seulgi Jung 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

3. 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

4. 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

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
Linear Programming 2015

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
Linear Programming 2015

7. 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

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 …
Linear Programming 2015

9. Resources Societies: Journals:
INFORMS (the Institute for Operations Research and Management Sciences) :
MOS (Mathematical Optimization Society) :
Korean Institute of Industrial Engineers :
Korean Operations Research Society :
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

10. 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

11. 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

12. ∃ 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