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final250 - UNIVERSITY OF WATERLOO FINAL EXAMINATION FALL...

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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 Total 110 (Full credit = 100)
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2 Question 1 (15 marks) . Matching algorithm Consider the following bipartite graph with edge weights. 1 4 4 0 -1 2 0 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 4 0 -1 2 0 1 4 4 0 -1 2 0 1 4 4 0 -1 2 0 1 4 4 0 -1 2 0
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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 0 . (i) Find two extreme points of F . Justify your answer.
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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 Which rows of this tableau lead to cutting-planes based on our construction in class? From each such row, generate a cutting-plane for the underlying IP
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