Kungl Tekniska högskolan / Optimization and Systems Theory /

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SF2812 Applied Linear Optimization, 7.5hp, 2016/2017

Instructor and examiner

Anders Forsgren (andersf@kth.se), room 3533, Lindstedtsv. 25, tel 790 71 27.
Office hours: Monday 11-12. (Or by agreement.)

Exercise leader and project leader

David Ek (daviek@kth.se), room 3736, Lindstedtsv. 25, tel. 790 62 94.
Office hours: By agreement.

Course material

  • Linear and Nonlinear Optimization, second edition, by I. Griva, S. G. Nash och A. Sofer, SIAM, 2009.
    (The book can be ordered from several places. Please note that you can become a SIAM member for free and obtain a discount at the SIAM bookstore.)
  • Exercises in applied linear optimization, 2016/2017. Available via Canvas.
  • Lecture notes in applied linear optimization, 2016/2017. May be downloaded from this web page, see the schedule below. Also available via Canvas.
  • Supplementary course material in applied linear optimization, 2016/2017. Available via Canvas.
  • Theory questions in applied linear optimization, 2016/2017. Available via Canvas.
  • GAMS, A user's guide. May be downloaded from the GAMS web site.
  • GAMS. GAMS is installed in the KTH linux computer rooms. It may also be downloaded from the GAMS web site for use on a personal computer.
  • Two project assignments that are handed out during the course, February 2 and February 16 respectively.

Additional notes that may be handed out during the course are also included.

Course goals

After completed course, the student should be able to:
  • explain fundamental concepts of linear programming and integer linear programming;
  • explain how fundamental methods for linear programming and integer linear programming work;
  • illustrate how these methods work by solving small problems by hand calculations;
  • starting from a suitably modified real problem, formulate a linear program or an integer linear program; make a model in a modeling language and solve the problem;
  • analyze the solutions of the optimization problem solved, and present the analysis in writing as well as orally;
  • interact with other students when modeling and analyzing the optimization problems.


The examination is in two parts, projects and final exam. To pass the course, the following requirements must be fulfilled:
  • Pass project assignment 1, with presence at the compulsory presentation lecture on Thursday February 16, and presence at the following dicussion session.
  • Pass project assignment 2, with presence at the compulsory presentation lecture on Thursday March 2 and presence at the following dicussion session.
  • Pass final exam.

Course registration

Due to the project based nature of this course, students must register no later than January 31. Registration is made by the students online following KTH standard procedures. PhD students register via e-mail to the instructor.

Project assignments

The project assignments are performed in groups, where the instructor determines the division of groups. This division is changed between the two assignments. The assignments are carried out by the modeling language GAMS. The project assignments must be carried out during the duration of the course and completed by the above mentioned presentation lectures. Presence at the presentation lectures is compulsory. For passing the projects, the following requirements must be fulfilled:
  • No later than the night before the presentation lecture, each group must hand in a well-written report which describes the exercise and the group's suggestion for solving the exercise through Canvas as a pdf file. Suitable word processor should be used. The report should be on a level suitable for another participant in the course who is not familiar with the group's specific problem.
  • When handing in the report, each student should append an individual sheet with a brief self-assessment of his/her contribution to the project work, quantitatively as well as qualitatively.
  • At the presentation lecture, all assignments will be presented and discussed. Each student is expected to be able to present the assignment of his/her group, the modeling and the solution. In particular, each student is expected to take part in the discussion. The presentation and discussion should be on a level such that students having had the same assignment can discuss, and students not having had the same assignment can understand the issues that have arisen and how they have been solved.
  • Each group should make an appointment for a discussion session with the course leaders. There is no presentation at this session, but the course leaders will ask questions and give feedback. There will be time slots available the days after the presentation session. One week prior to the presentation lecture, a list of available times for discussion sessions will be made available at Doodle, reachable from the course home page. Each group should sign up for a discussion session prior to the presentation lecture.

Each project assignment is awarded a grade which is either fail or pass with grading E, D, C, B and A. Here, the mathematical treatment of the problem as well as the report and the oral presentation or discussion is taken into account. Normally, the same grade is given to all members of a group

Each group must solve their task independently. Discussion between the groups concerrning interpretation of statements etc. are encouraged, but each group must work independently without making use of solutions provided by others. All groups will not be assigned the same exercises.

Final exam

The final exam consists of five exercises and gives a maximum of 50 points. At the exam, the grades F, Fx, E, D, C, B and A are awarded. For a passing grade, normally at least 22 points are required. In addition to writing material, no other material is allowed at the exam. Normally, the grade limits are given by E (22-24), D (25-30), C (31-36), B (37-42) and A (43-50).

The grade Fx is normally given for 20 or 21 points on the final exam. An Fx grade may be converted to an E grade by a successful completion of two supplementary exercises, that the student must complete independently. One exercise among the theory exercises handed out during the course, and one exercise which is similar to one exercise of the exam. These exercises are selected by the instructor, individually for each student. Solutions have to be handed in to the instructor and also explained orally within three weeks of the date of notification of grades.

The final exam is given Monday March 13 2016, 8.00-13.00.

Final grade

By identitying A=7, B=6, C=5, D=4, E=3, the final grade is given as

round( (grade on proj 1) + (grade on proj 2) + 2 * (grade on final exam) ) / 4),

where the rounding is made to nearest larger integer in case of a tie.

Preliminary schedule

"L" means lecture, "E" means exercise session, "P" means project session.
Type Day Date Time Room Subject
L1.Tue Jan 17 15-17 L52 Introduction. Linear programming models. (pdf)
L2.Thu Jan 19 8-10 L51 Linear programming. Geometry. (pdf)
L3.Fri Jan 20 13-15 L51 Lagrangian relaxation. Duality. LP optimality. (pdf)
L4.Tue Jan 24 15-17 L52 Linear programming. The simplex method. (pdf)
E1.Thu Jan 26 8-10 L51 Linear programming. The simplex method.
L5.Fri Jan 27 13-15 L52 More on the simplex method. (pdf)
E2.Tue Jan 31 15-17 L51 Linear programming. The simplex method.
P1.Wed Feb 1 10-12 V34 Introduction to GAMS. (pdf)
P2.Thu Feb 2 8-10 Brun GAMS excercise session.
L6.Fri Feb 3 8-10 V34 Stochastic programming. (pdf)
E3.Tue Feb 7 15-17 L51 Stochastic programming.
L7.Thu Feb 9 8-10 L51 Interior methods for linear programming. (pdf)
E4.Fre Feb 10 8-10 E2 Interior methods for linear programming.
L8.Tue Feb 14 15-17 L52 Integer programming models. (pdf)
L9.Wed Feb 15 10-12 V34 Branch-and-bound. (pdf)
P3.Thu Feb 16 8-10 L41, L52 Presentation of project assignment 1.
E5.Fri Feb 17 8-10 L52 Integer programming.
L10.Tue Feb 21 15-17 Q34 Decomposition and column generation. (pdf)
E6.Thu Feb 23 8-10 Q31 Decomposition and column generation.
L11.Fri Feb 24 8-10 E2 Lagrangian relaxation. Duality. (pdf)
E7.Tue Feb 28 15-17 L52 Lagrangian relaxation. Duality.
P4.Thu Mar 2 8-10 V12, V32 Presentation of project assignment 2.
L12.Fri Mar 3 8-10 E2 Subgradient methods. (pdf)
E8.Mon Mar 6 10-12 L51 Subgradient methods.

Mapping of exercises to lectures

The sections in the exercise booklet may roughly be mapped to the lectures as follows:
  • 1. The simplex method. After L4.
  • 2. Sensitivity analysis. After L4.
  • 3. Interior point methods. After L7.
  • 4. Decomposition and column generation. After L10.
  • 5. Linear programming - remaining. After L7.
  • 6. Stochastic programming. After L6.
  • 7. Formulation - integer programming. After L8.
  • 8. Lagrangian relaxation and duality. After L11.
  • 9. Subgradient methods. After L12.

Overview of course contents

  • Linear programming
    Fundamental LP theory with corresponding geometric interpretations. The simplex method. Column generation. Decomposition. Duality. Complementarity. Sensitivity. Formulations of LPs. Interior methods for linear programming, primal-dual interior methods in particular.
    (Chapters 4-7 in Griva, Nash and Sofer, except 5.2.3, 5.2.4, 5.5.1, 6.5, 7.5, 7.6. Chapter 9.3 in Griva, Nash and Sofer. Chapter 10 in Griva, Nash and Sofer, except 10.3, 10.5.)
  • Stochastic programming
    Fundamental theory. (Supplementary course material.)
  • Integer programming
    Formulations of integer programs. Branch-and-bound. Lagrangian relaxation and subgradient methods applied on integer programs with special structure.
    (Supplementary course material.)

Welcome to the course!

Course home page: http://www.math.kth.se/optsyst/grundutbildning/kurser/SF2812/.

Optimization and Systems Theory, KTH
Anders Forsgren, andersf@kth.se