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finalS09 - 1 UNIVERSITY OF WATERLOO FINAL EXAMINATION...

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Unformatted text preview: 1 UNIVERSITY OF WATERLOO FINAL EXAMINATION SPRING TERM 2009 Surname: First Name: Name:(Signature) Id.#: Section #: Course Number CO 350 Course Title Linear Optimization Instructors L. Tun¸ cel (Section 1), E. Teske (Section 2) Date of Exam August 14, 2009 Time Period 4:00 pm - 6:30 pm Number of Exam Pages 13 (including this cover sheet) Exam Type Closed Book Additional Materials none Additional Instructions 1. Write your answers in the space provided. 2. You may use the backs of the pages if necessary. 3. Calculators are not permitted. 4. Some questions require you to prove or disprove certain statements. In doing so, you may refer to any result from the lecture notes except the particular one you are trying to prove or disprove. You must write down the statement of the result you are using clearly and completely. Problem Value Mark Awarded Problem Value Mark Awarded 1 15 6 10 2 20 7 15 3 10 8 10 4 10 9 10 5 20 Total---- 2 1. [15 points] [5 points] (a) Write down the dual of the following linear programming problem: maximize c T x + d T u subject to Ax + Du = b x ≥ , where A ∈ R m × n , D ∈ R m × p , b ∈ R m , c ∈ R n and d ∈ R p all given as data; x ∈ R n and u ∈ R p are the variable vectors. [5 points] (b) State Farkas’ Lemma. [5 points] (c) Let A be m by n . Prove that the system Ax ≥ b, x ≥ has no solution if and only if there exists y ∈ R m such that y ≥ , A T y ≤ , b T y > . 3 2. [20 points] Let A := − 2 1 − 3 1 2 − 1 2 0 2 − 1 1 0 , b := 3 2 4 , c := [1 , 1 , , 2] T define the data for an LP problem in standard equality form. [5 points] (a) Introduce artificial variables x 5 and x 6 , and write down an auxiliary problem ( Aux ) whose solution determines whether the original LP problem has feasible solution(s). [10 points] (b) Solve the auxiliary problem ( Aux ) using the Revised Simplex Method with the smallest subscript rule for choosing entering and leaving variables. [5 points] (c) Use the final optimal basis from Phase-I to either make a feasible Simplex Tableau for the original LP problem, or give an algebraic proof that the original LP problem is infeasible. 4 3. [10 points] Consider the following tableau. Assume that the initial basis was B = { 5 , 6 , 7 } with A B = I ....
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finalS09 - 1 UNIVERSITY OF WATERLOO FINAL EXAMINATION...

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