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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 Notation 2

  3. Course Organization Outline Preliminaries 1. Course Organization 2. Preliminaries Notation 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 � You will apply the tools learned to solve real life problems using computer software 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 Rank, range and linear equations 5 Vector spaces 6 Linear Transformations - Matrix representation 7 Diagonalization - Eigenvalues and Eigenvectors 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 (Hold H1): Lasse Malm Lidegaard Instructor (Hold H2): Marco Chiarandini 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 immediately when you think we should do things differently! 8

  9. Course Organization Books Preliminaries Linear Algebra Part : Le Steven J. Leon, Linear Algebra with Applications, 8th edition, Prentice Hall (2010). Other books: AH Martin Anthony and Michele Harvey, Linear Algebra, Concepts and Methods. 2012. Cambridge AR Howard Anton and Chris Rorres. Elementary Linear Algebra. 11th Edition. 2015. Wiley 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: 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 • Three obligatory Assignments, pass/fail, evaluation by teacher practical exercises modeling + programming • 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 • Best in groups of 2 • They require the use of Python + a MILP Solver (2nd part) Software available for all systems from the Main 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 Who is here? Preliminaries 60 registered in BlackBoard... • Computer Science (2nd year, 4th semester) • Applied Mathematics? • Math-economy? Prerequisites • Calculus (MM501, MM502) 15

  16. Course Organization Previous evaluation Preliminaries The majority of the students finds: • indifferent the quality of the text books and exercises in class while satisfactory the lecture notes and the slides. • satisfactory the preparation of the teacher while indifferent or dissatisfactory his pedagogical competencies. "M seems annoyed when asking something “trivial”" • Marco as instructor has too high expectations and no understanding of where the students are. • in general the course was pleasant and intellectually stimulating • the volume of work necessary to complete this course means that it cannot all be thoroughly comprehended. • there were too many exercises for the exercise sessions. The language in the exercises contains heavy mathematical notation and it is not easy to understand. • motivation and goals were made clear • the obligatory assignments were difficult • the review process in Assignment 1 was positive • low attendance was due to i) several assignments during the semester ii) the lagging behind due to lack of attendance in exercise sessions 16

  17. Course Organization Coding Preliminaries • gives you the ability to create new and useful artifacts with just your mind and your fingers, • allows you to have more control of your world as more and more of it becomes digital, • is just fun. It can also help you understand math! Being able to turn procedural ideas into code and run the code on concrete examples gives you a great advantage in developing and reinforcing your understanding of mathematical concepts. Beside: • listening to lectures • watching an instructor work through a derivation • working through numerical examples by hand you can learn by doing interacting with Python. • Python 2.7 or 3.4? • ipython (= interactive python)? 17

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

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

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

  21. 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 } 21

  22. 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) 22

  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 − 1 = 1 /a n , a 1 /n = x ⇐ ⇒ x n = a , a m/n = ( a 1 /n ) m 23

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