Lecture Summaries for COMP 566A
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Linear Programming, V. Chvatal
Sept 2: Introduction to Mathematical programming,
Operations research and course goals. Formulations of "the tragedy of the commons"
Sept 7: How to solve LPs in standard form. Read
ch 2 of the text. Make up and solve one complete example by hand !
Sept 9: The simplex method in terms of dictionaries.
Dantzig and Bland's pivot selection rule. Examples of cycling. Read
Ch. 3 up to p.38.
treatment of the lexicographic ratio test is contained in Ignizio
& Cavalier, Linear Programming, pp118-122, on reserve in Schulich
Sept 14: Geometry of the simplex method. Degeneracy
in polyhedra. Klee -Minty examples of worst case behaviour for Dantzig's
text: Ch. 4 and pp. 250-260.
Sept 16: Initialization of the simplex method.
Introduction to duality. Read Remainder of Ch. 3 and pp 54-57.
way of getting a feasible solution to a system of inequalities is described
Sept 21: Duality theory. Proofs of the weak and strong duality theorems.
Read: Ch 5 up to p. 62.
Sept 23: Complementary slackness. Economic interpretations
of duality. Proof of the finiteness of Bland's rule. Read: Rest
of Ch 5. Also pp. 37-8 Thm 3.3
Sept 28: Introduction to Revised Simplex method.
How to avoid taking the inverse of the basis matrix.
Sept 30: An example of Revised Simplex. Eta factorization
of the basis matrix. Deriving Farkas lemma from phase 1 of the Simplex
Method. Read: Ch. 7 and pp143-5.
Oct 5: General LP problems: how to handle bounded
variables. Duality for LPs not in standard form. Read: Ch 8, pp118-124
and C h. 9 (except proof of duality theorem).
Oct 7: Review. Test will cover all of the above
Oct 12: First class test.
Oct 14: Review of test. The dual simplex method.
Introduction to integer programming. Read: Text, Ch 10 up
Oct 19: Gomory's cutting plane algorithm for integer
programming. Read: class handout.
Oct 21: Zero sum games. LP formulation and minmax
theorem. Read: Text, Chapter 15
Oct 26: Non-zero sum games and Nash equilibria.
Read: First 10 pages of von Stengel's survey here.
Supplemental notes here.
Oct 28: Basic information on vertex enumeration
can be found in text pp. 271-275. The approach given in class is based
on my paper
(Read up to the end of Prop. 3.1.).
A tutorial on reverse
search is contained in the paper.
(will not be covered in test.)
Nov 2,4: Introduction to primal dual interior point
methods. Read: Class handout. Second class test on Nov 11 covers all material
up to Nov 4 lecture.
Nov 9: Review, Nash games again.
Nov 11: Second Class test.
Nov 16, 18: More on interior point methods: convergence,
feasibility regions, barrier methods.
Nov 23, 25, 30: Class presentations. Attendance
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