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Unformatted text preview: UNIVERSITY OF WATERLOO FINAL EXAMINATION FALL TERM 2010 Surname: First Name: Signature: Id.#: Section#: Course Number CO250/CM340 Course Title Introduction to Optimization Instructors B. Guenin, L. Tun cel Date of Exam December 15th, 2010 Time Period 7:30 PM  10:00 PM Number of Exam Pages (including cover page) 14 pages Exam Type Closed Book Additional Materials Allowed none Problem Value Mark Awarded 1 15 2 15 3 15 4 15 5 15 6 10 7 10 8 15 2 Question 1 (15 marks) . Matching algorithm Consider the following bipartite graph with edge weights. 1 4 41 2 Using the algorithm seen in the course, find a maximum weight perfect matching. Note, I have reproduced several copies of the graph below for your convenience. Make sure to indicate the y values, the deficient sets, and the equality edges at every step. 1 4 41 2 1 4 41 2 1 4 41 2 1 4 41 2 3 Question 2 (15 marks) . Extreme points, Basic feasible solutions (a) Let C R n be a convex set. Define what it means for x R n to be an extreme point of C . (b) Let A R m n such that rank( A ) = m . Define what it means for B { 1 , 2 ,...,n } to be a basis of A . (c) Let F be the set of x satisfying x 1 + x 2 541 x 3 + x 4 = 3 2 x 1 + 229 x 3 2 x 4 = 2 x 1 , x 2 , x 3 , x 4 . (i) Find two extreme points of F . Justify your answer. 4 Question 3 (15 marks) . Integer programming tools (a) Suppose that the optimal Simplex tableau for the LP relaxation of an IP prob lem where all the variables are required to be integer is as follows, z + 17 12 x 4 + 1 12 x 5 + 5 12 x 6 = 140 x 1 + 111 60 x 4 13 12 x 5 1 60 x 6 = 9 2 x 2 + 1 10 x 4 + 1 2 x 5 1 10 x 6 = 1 x 3 +2 x 4 + 11 2 x 5 1 12 x 6 = 11 10...
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This note was uploaded on 06/12/2011 for the course CO 250 taught by Professor Guenin during the Spring '10 term at Waterloo.
 Spring '10
 GUENIN

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