The course deals with theory and algorithms for linear programming problems.

From the 1940s the simplex method, developed by Dantzig, was the only practically used method for solving linear programming problems. Khachian had in the late 1970s presented the polynomial ellipsoid method, but it had not been successful in practice.

When Karmarkar presented his interior method in 1984, all this changed. This method was polynomial and also claimed to be superior to the simplex method in practice.

Karmarkar's method lead to an "explosion" within the area of linear programming. Gill et. al. soon showed that Karmarkar's method was equivalent to a logarithmic barrier method, and the development of new interior methods was rapid.  This "competition" between the simplex method and interior methods has lead to significant improvement in both types of method. The purpose of this course is to reflect this development. Some more advanced aspects of the simplex method are included, e.g., steepest edge, partial pricing, and of the interior-point methods e.g., predictor-corrector methods. In particular, we try to understand how the different methods work.

That the student should obtain a deep understanding of the mathematical theory and the numerical methods for linear programming.

After completed course, the student should be able to

• Derive fundamental concepts related to polyhedrals of linear programs
• Explain fundamental duality concepts for linear programming.
• Explain how the simplex method works, primal simplex, dual simplex, steepest edge.
• Explain how interior methods work, in particular primal-dual methods

The course consists of 24h lectures, given during periods 1 and 2, autumn 2022.

Lectures will be given in Room 3721, Lindstedtsvägen 25.

There will be five sets of homeworks and an oral final exam. Homework assignment and other material related to the course will be posted in Canvas.

The literature is a textbook, a set of articles and extract from textbooks. Below is a listing of these articles and books, where it is also indicated what parts are of significant importance.

Students are expected to have access to the textbook [15]. The book can for example be ordered directly from SIAM or you can access the book online via KTH. (KTH PhD students can become SIAM members for free, as KTH is an academic member of SIAM.)

1. V. Chvátal. Linear Programming. W. H. Freeman and Company, New York, 1983. (Chapters 24 and 25 available in Canvas).
2. J. J. Forrest and D. Goldfarb. Steepest-edge simplex algorithms for linear programming. Mathematical Programming 57 (1992), 341-374.
3. A. Forsgren, An elementary proof of optimality conditions for linear programming. Report TRITA-MAT-2008-OS6, Department of Mathematics, Royal Institute of Technology, 2008.
4. A. Forsgren, P. E. Gill and M. H. Wright, Interior methods for nonlinear optimization. SIAM Review 44 (2002), 525-597. (Basic ideas.)
5. A. Forsgren, P. E. Gill and E. Wong, Primal and dual active-set methods for convex quadratic programming, Mathematical Programming 159 (2016), 469-508. (For the linear programming case.)
6. A. Forsgren and F. Wang, On the existence of a short pivoting sequence for a linear program. Operations Research Letters 48 (2020), 697-702.
7. P. E. Gill, W. Murray, M. A. Saunders, J. Tomlin, and M. H. Wright. On projected Newton methods for linear programming and an equivalence to Karmarkar's projective method. Mathematical Programming 36 (1986), 183-209.
8. P. E. Gill, W. Murray, and M. H. Wright. Numerical Linear Algebra and Optimization, volume 1. Addison-Wesley Publishing Company, Redwood City, 1991. (Reference book on linear programming. Not required.)
9. D. Goldfarb and J. K. Reid. A practicable steepest-edge simplex algorithm. Mathematical Programming 12 (1977), 361-371.
10. D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Handbooks in Operations Research and Management Science, volume 1. Optimization, chapter 2, pages 73-170. North Holland, Amsterdam, New York, Oxford and Tokyo, 1989. (Reference article on linear programming. Not required.)
11. N. Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica 4 (1984), 373-395. (Basic ideas.)
12. S. Mehrotra. On finding a vertex solution using interior point methods. Linear Algebra and its Applications 152 (1991), 233-253. (Section 4, basic ideas.)
13. S. Mehrotra. On the implementation of a primal-dual method. SIAM Journal on Optimization 2 (1992), 575-601. (Sections 1 and 2, basic ideas.)
14. I. Griva, S. G. Nash and A. Sofer, Linear and nonlinear programming, SIAM, 2009. ISBN: 978-0-898716-61-0. (Reference book on linear programming. Not required.)
15. S. J. Wright. Primal-dual interior-point methods. SIAM, Philadelphia, USA, 1997. ISBN 0-89871-382-X.

Students at KTH with a permanent disability can get support during studies from Funka:

Funka - compensatory support for students with disabilities

P, F

• INL1 - Assignment, 7.5 credits, Grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The examination is by homework assignments and a final oral exam.

Homework assignments and a final oral exam.

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.