Math 5593 Linear Programming Final Exam
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1 Math 5593 Linear Programming Final Exam Department of Mathematical and Statistical Sciences University of Colorado Denver, Fall 2013 Name: Points: /30 This exam consists of 6 problems, and each problem is worth 5 points for a total of 30 points. You are not allowed to use your books, notes, or any other auxiliary materials on this exam. If you are asked to give a proof, you must use full sentences and correct mathematical notation. If you are asked to give a model, you must clearly define all sets, variables, and parameters. Problem 1: State, prove, and give a geometric interpretation of the Duality Theorem of Linear Programming. In your proof, you may use any other results from class without proving them again. Problem 2: After returning from her semester in Beijing, Jenny will be looking for a new place to live. She prefers to live close to the university for work, and to City Park and Cheesman Park for running or visits to Zoo, Science Museum, or Botanic Gardens. Looking at a Downtown Denver map, she estimates coordinates (2,3) for Auraria Campus, (5,1) for Cheesman Park, and (6,4) for City Park. Assuming that travel will only occur into north-south and east-west direction, she can compute her total travel distance as the sum of movements in the vertical and horizontal directions (for example, the distance between Auraria Campus and Cheesman Park is = 5). (a) Formulate an LP that Jenny could use to find a place with the smallest total travel distance to Auraria Campus, Cheesman Park, and City Park. Remember to linearize any absolute values. (b) Find the optimal solution to your model (no optimization or computations, please!) and briefly explain why the corresponding location is indeed optimal (use a basic concept from statistics). Problem 3: Let (c, a, b) Z n Z n Z be given and consider the equality knapsack problem: maximize c T x subject to a T x = b and x Z n +. (a) Formulate its linear programming relaxation. (b) Derive conditions on c, a, and b under which the relaxation is feasible and bounded. (c) Do these conditions also imply that the integer problem is feasible and bounded? Explain. (d) Assuming that the relaxation is both feasible and bounded, characterize its optimal solution(s). (e) Give the meaning of the value of the dual variable associated with the (continuous) knapsack constraint and explain why this meaning does not translate to the original (discrete) problem. Problem 4: Solve the integer rolling mill problem using branch and bound. You may draw the feasible region to solve each subproblem by inspection, but you should properly fathom your resulting search tree: for each node/formulation, indicate its new inequality, its solution, and its new bound. Problem 5: This problem has three parts. First, list the four major steps common to (almost) all algorithms that solve continuous optimization problems. Second, for linear programs specifically, explain the basic idea and differences of the three algorithms (simplex, affine-scaling, path-following) discussed in class. Third, also address some of the advantages and disadvantages of each method. Problem 6: The following article was published in The New York Times on November 27, 1979 as a reaction to Khachiyan s development of the ellipsoid method for linear programming. Edit this article by marking (e.g., underline or cross out) ambiguous or wrong statements, and suggest possible clarifications or corrections (e.g., use number labels and write your new statements below).
2 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 1: State, prove, and give a geometric interpretation of the Duality Theorem of Linear Programming. In your proof, you may use any other results from class without proving them again.
3 Math 5593 Linear Programming Final Exam, UC Denver, Fall
4 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 2: After returning from her semester in Beijing, Jenny will be looking for a new place to live. She prefers to live close to the university for work, and to City Park and Cheesman Park for running or visits to Zoo, Science Museum, or Botanic Gardens. Looking at a Downtown Denver map, she estimates coordinates (2,3) for Auraria Campus, (5,1) for Cheesman Park, and (6,4) for City Park. Assuming that travel will only occur into north-south and east-west direction, she can compute her total travel distance as the sum of movements in the vertical and horizontal directions (for example, the distance between Auraria Campus and Cheesman Park is = 5). (a) Formulate an LP that Jenny could use to find a place with the smallest total travel distance to Auraria Campus, Cheesman Park, and City Park. Remember to linearize any absolute values. (b) Find the optimal solution to your model (no optimization or computations, please!) and briefly explain why the corresponding location is indeed optimal (use a basic concept from statistics)
5 Math 5593 Linear Programming Final Exam, UC Denver, Fall
6 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 3: Let (c, a, b) Z n Z n Z be given and consider the equality knapsack problem: maximize c T x subject to a T x = b and x Z n +. (a) Formulate its linear programming relaxation. (b) Derive conditions on c, a, and b under which the relaxation is feasible and bounded. (c) Do these conditions also imply that the integer problem is feasible and bounded? Explain. (d) Assuming that the relaxation is both feasible and bounded, characterize its optimal solution(s). (e) Give the meaning of the value of the dual variable associated with the (continuous) knapsack constraint and explain why this meaning does not translate to the original (discrete) problem.
7 Math 5593 Linear Programming Final Exam, UC Denver, Fall
8 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 4: Solve the integer rolling mill problem using branch and bound. You may draw the feasible region to solve each subproblem by inspection, but you should properly fathom your resulting search tree: for each node/formulation, indicate its new inequality, its solution, and its new bound. maximize 5x 1 + 6x 2 subject to 7x x 2 56 x 1 6 x 2 4 x 1, x 2 0 integer
9 Math 5593 Linear Programming Final Exam, UC Denver, Fall
10 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 5: This problem has three parts. First, list the four major steps common to (almost) all algorithms that solve continuous optimization problems. Second, for linear programs specifically, explain the basic idea and differences of the three algorithms (simplex, affine-scaling, path-following) discussed in class. Third, also address some of the advantages and disadvantages of each method.
11 Math 5593 Linear Programming Final Exam, UC Denver, Fall
12 Math 5593 Linear Programming Final Exam, UC Denver, Fall Problem 6: The following article was published in The New York Times on November 27, 1979 as a reaction to Khachiyan s development of the ellipsoid method for linear programming. Edit this article by marking (e.g., underline or cross out) ambiguous or wrong statements, and suggest possible clarifications or corrections (e.g., use number labels and write your new statements below)
13 Math 5593 Linear Programming Final Exam, UC Denver, Fall
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