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DM559 Linear and Integer Programming Lecture 1 Introduction Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Course Organization Outline Preliminaries 1. Course Organization 2.

  1. DM559 Linear and Integer Programming Lecture 1 Introduction Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

  2. Course Organization Outline Preliminaries 1. Course Organization 2. Preliminaries 2

  3. Course Organization Outline Preliminaries 1. Course Organization 2. Preliminaries 3

  4. Course Organization Aims of the course Preliminaries Learn about: • both the theory and the practice of Linear Algebra • one of the most important applications of Linear Algebra: • Mathematical optimization: linear programming • Discrete optimization: integer programming � Practical experience with computer software � The solution to really many real-life problems will be within short reach 4

  5. Course Organization Contents of the Course (1/2) Preliminaries (see Syllabus) Linear Algebra: manipulation of matrices and vectors with some theoretical background Linear Algebra 1 Matrices and vectors - Matrix algebra, Geometric insight 2 Systems of Linear Equations - Gaussian elimination 3 Matrix inversion and determinants 4 Vector spaces 5 Linear Transformations - Matrix representation 6 Diagonalization - Eigenvalues and Eigenvectors (7 Orthogonality or Numerical Methods) 5

  6. Course Organization Contents of the Course (2/2) Preliminaries (see Syllabus) Linear Programming 1 Introduction - Linear Programming, Notation 2 Linear Programming, Simplex Method 3 Exception Handling 4 Duality Theory 5 Sensitivity 6 Revised Simplex Method Integer Linear Programming 7 Modeling Examples, Good Formulations, Relaxations 8 Well Solved Problems 9 Network Optimization Models (Max Flow, Min cost flow, matching) 10 Cutting Planes & Branch and Bound 11 More on Modeling 6

  7. Course Organization Practical Information Preliminaries Teacher: Marco Chiarandini ( http://www.imada.sdu.dk/~marco ) Instructor: Jens Østergaard Schedule: • Introductory classes: 44 hours (22 classes) • Training classes: 50 hours - Exercises: 42 hours - Laboratory: 8 hours Alternative views of the schedule: • http://mitsdu.sdu.dk/skema , SDU Mobile • Official course description (læserplaner) • http://www.imada.sdu.dk/~marco/DM559 7

  8. Course Organization Communication Means Preliminaries • BlackBoard (BB) ⇔ Main Web Page (WWW) (link http://www.imada.sdu.dk/~marco/DM559 ) • Announcements in BlackBoard • Discussion Board in (BB) - allowed anonymous posting and rating • Write to Marco (marco@imada.sdu.dk) and to instructor • Ask peers • You are welcome to visit me in my office in working hours (8-16) � It is good to ask questions!! � Let me know when you think we should do things differently! Things can be changed. 8

  9. Course Organization Books Preliminaries Linear Algebra Part : AR Howard Anton and Chris Rorres. Elementary Linear Algebra. 11th Edition. 2014. Wiley Other books: AH Martin Anthony and Michele Harvey, Linear Algebra, Concepts and Methods. 2012. Cambridge Le Steven J. Leon, Linear Algebra with Applications, 8th edition, Prentice Hall (2010). FSV Computing with Python: An introduction to Python for science and engineering Claus Führer, Jan Erik Solem, Olivier Verdier 9

  10. Course Organization Books Preliminaries Linear and Integer Programming Part: LN Lecture Notes MG J. Matousek and B. Gartner. Understanding and Using Linear Programming. Springer Berlin Heidelberg, 2007 Wo L.A. Wolsey. Integer programming. John Wiley & Sons, New York, USA, 1998 Other books: HL Frederick S Hillier and Gerald J Lieberman, Introduction to Operations Research, 9th edition, 2010 • ... 10

  11. Course Organization Course Material Preliminaries Main Web Page (WWW) is the main reference for list of contents (ie 1 , syllabus, pensum). It contains: • slides • list of topics and references • exercises • links • software 1 ie = id est, eg = exempli gratia, wrt = with respect to 11

  12. Course Organization Assessment Preliminaries • 7.5 ECTS • (Four) Obligatory Assignments, pass/fail, evaluation by teacher both theoretical and practical (programming) exercises • 4 hour written exam, 7-grade scale, external censor (theory part) similar to exercises in class and past exams in June • (language: Danish and/or English) 12

  13. Course Organization Obligatory Assignments Preliminaries • Small practical tasks must be passed to attend the written exam • Individual work • They require the use of Python + a MILP Solver (2nd part) See Tools from Public Web Page 13

  14. Course Organization Training Sessions Preliminaries • Prepare them in advance to get out the most • Best carried out in small groups • Exam rehearsal (in June?) 14

  15. Course Organization Some Numbers Preliminaries DM559 DM545 Obligatory Assignments Obligatory Assignments In Blackboard 60 In Blackboard 60 Registered 43 Registered 27 Submitted 24 Submitted 19 Passed 19 Passed 18 Final exam: Final exam: Registered 19 Registered 29 Showed up 14 Showed up 21 Passed 6 Passed 16 Reexam: Reexam: Registered 12 Registered 12 Showed up 6 Showed up 4 Passed 4 Passed 4 15

  16. Course Organization Who is here? Preliminaries 65 registered in BlackBoard... • Computer Science (2nd year, 4th semester) • Applied Mathematics? • Math-economy? • Returning CS Prerequisites • Goodwill • Algorithms and Data Structures, Programming 16

  17. Course Organization Coding Preliminaries Coding smal examples can help you to understand math! Beside: • listening to lectures • watching an instructor work through a derivation • working through numerical examples by hand you can learn by doing by interacting with Python. • jupyter.org/ipython (= interactive python) • Python 2.7 or 3.5? 19

  18. Course Organization Outline Preliminaries 1. Course Organization 2. Preliminaries 20

  19. Course Organization Overview Preliminaries • Notation • Martices and vectors: • matrix arithmetic operations (addition, subtraction, and multiplication) • scalar multiplication and transposition. 21

  20. Course Organization Sets Preliminaries • A set is a collection of objects. eg.: A = { 1 , 2 , 3 } • A = { n | n is a whole number and 1 ≤ n ≤ 3 } (’ | ’ reads ’such that’) • B = { x | x is a reader of this book } • x ∈ A x belongs to A • set of no members: empty set, denoted ∅ • if a set S is a (proper) subset of a set T , we write ( S ⊂ T ) T ⊆ S { 1 , 2 , 5 } ⊂ { 1 , 2 , 4 , 5 , 6 , 30 } • for two sets A and B , the union A ∪ B is { x | x ∈ A or x ∈ B } • for two sets A and B , the intersection A ∩ B is { x | x ∈ A and x ∈ B } { 1 , 2 , 3 , 5 } and B = { 2 , 4 , 5 , 7 } , then A ∩ B = { 2 , 5 } 23

  21. Course Organization Numbers Preliminaries • set of real numbers: R • set of natural numbers: N = { 1 , 2 , 3 , 4 , ... } (positive integers) • set of all integers: Z = { ..., − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , ... } • set of rational numbers: Q = { p/q | p, q ∈ Z , q � = 0 } • set of complex numbers: C • absolute value (non-negative): � a if a ≥ 0 | a | = − a if a ≤ 0 | a + b | ≤ | a | + | b | , a, b ∈ R • the set R 2 is the set of ordered pairs ( x, y ) of real numbers (eg, coordinates of a point wrt a pair of axes, the Cartesian plane) 24

  22. Course Organization Matrices and Vectors Preliminaries • A matrix is a rectangular array of numbers or symbols. It can be written as   a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n   . . .  ...  . . .   . . .   a m 1 a m 2 · · · a mn • An n × 1 matrix is a column vector, or simply a vector:  v 1  v 2   v =  .  .   .   v n • the set R n is the set of vectors [ x 1 , x 2 , . . . , x n ] T of real numbers (eg, coordinates of a point wrt an n -dimensional space, the Euclidean Space) 25

  23. Course Organization Basic Algebra Preliminaries Elementary Algebra: the study of symbols and the rules for manipulating symbols. It differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values • collecting up terms: eg. 2 a + 3 b − a + 5 b = a + 8 b • multiplication of variables: eg: a ( − b ) − 3 ab + ( − 2 a )( − 4 b ) = − ab − 3 ab + 8 ab = 4 ab • expansion of bracketed terms: eg: − ( a − 2 b ) = − a + 2 b, (2 x − 3 y )( x + 4 y ) = 2 x 2 − 3 xy + 8 xy − 12 y 2 = 2 x 2 + 5 xy − 12 y 2 • a r a s = a r + s , ( a r ) s = a rs , a − n = 1 /a n , a 1 /n = x ⇐ ⇒ x n = a , a m/n = ( a 1 /n ) m 26

  24. Course Organization Variables Preliminaries • In Mathematics and Statistics, a variable is an alphabetic character representing a value, which is unknown. They are used in symbolic calculations. Commonly given one-character names. • in contrast, a constant or given or scalar is a known real number • In Computer Science, a variable is a storage location paired with an associated identifier, which contains a value, which may be known or unknown. Commonly given long, explanatory names. 27

  25. Course Organization Functions Preliminaries • a function f on a set X into a set Y is a rule that assigns a unique element f ( x ) in S to each element x in X . y = f ( x ) y dependent x independent variable variable • a linear function has only sums and scalar multiplications, that is, for variable x ∈ R and scalars a, b ∈ R : f ( x ) := ax + b 28

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