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IE 426– Quiz #1 Answers 1 Short(ish) Answer 1.1 Easy or Hard Note: In this subsection, I am NOT asking you to solve the problem. Rather, I am asking you to say whether, given the “shape” of the objective function (over the feasible region) and the “shape” of the feasible region itself, would you expect an optimization solver to have an “easy” time solving this instance or a “hard” time solving this instance? You should state why you come to your conclusion. For example, your answers should all look something like the following: The objective function is concave , and the feasible region is convex , so the problem is XXX . (Where “XXX” is easy or hard ). 1.1 Problem (2 points) min x 2 s.t. x 0 x 14 . 5 x R Answer: Objective function is convex, feasible region is convex. Minimizing is Easy. 1.2 Problem (2 points) min 6 x 2 2 - 2 x 1 x 2 + 2 x 2 1 s.t. x 1 - x 2 15 x 1 , x 2 Z 2 + Answer: Objective function is convex, feasible region is not convex. Hard. 1.3 Problem (2 points) min 6 x 2 - 2 x 1 x 2 + 2 x 2 1 s.t. x 1 - x 2 15 x 1 , x 2 R 2 + Answer: Objective function is convex, feasible region is convex. Minimizing is Easy.

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IE426 Quiz #1 Name: 1.4 Problem (2 points) max 6 x 1 + 2 x 2 + 3 x 3 s.t. x 2 1 + x 2 2 4 x 3 8 x 1 , x 2 , x 3 R + Answer: Objective function is concave (and convex), feasible region is convex. Maximizing is Easy. Answer. 1.5 Problem (2 points) For parameters c R n , b R m , and A R m × n max c T x s.t. Ax b x u x ` Answer: Linear Programs are Easy. 1.2 What is “Wrong” With This Picture? (9) Consider the linear program: min c 1 x 1 + c 2 x 2 subject to 2 x 1 + 2 x 2 10 x 1 0 x 2 0 whose feasible region is graphed for you here: Problem 1 Page 2
IE426 Quiz #1 Name: X 1 X 2 Determine which of the following cases: A LP has Unique Optimal Solution B LP has Multiple Optimal Solutions C LP is Unbounded D None of the above best describes the linear program for the objective function stated in each problem. 1.6 Problem (2 points) c 1 = 0 , c 2 = - 1 Answer: C. Unbounded 1.7 Problem (2 points) c 1 = 2 , c 2 = 1 Answer: A. Unique Optimum. 1.8 Problem (2 points) c 1 = 1 , c 2 = 1 Answer: B. Multiple Optima. 1.3 Short Answer 1.9 Problem (3 points) A constraint in a linear program you created limits the total number of drilling hours available. Its algebraic form looks as follows: 10 x 1 + 8 x 2 + 16 x 3 192 . (1) The optimal solution to the linear program has x 1 = 10 , x 2 = 0 , x 3 = 4 , x 4 = 25 , x 5 = 0 . What is the optimal value for the dual variable to equation (1)? If it is impossible to determine from the given information, write impossible .

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