hw1_sol-1

hw1_sol-1 - MIT 2.098/6.255/15.093 Optimization Methods...

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Unformatted text preview: MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 1. Solution Problem 1: BT, exercise 1.4 Solution: One way to reformulate it as a linear programming problem is as follows: minimize 2 x 1 + 3 z subject to x 1 + 2 + x 2 5 x 1 + 2- x 2 5- x 1- 2 + x 2 5- x 1- 2- x 2 5 x 2- 10 z- x 2 + 10 z, here we use the fact | x 2- 10 | z x 2- 10 z and- x 2 + 10 z . Also the constraint inequality with two absolute values is broken into all four possible combinations. Another equivalent way is: minimize 2 x 1 + 3( z + + z- ) subject to x 1 + 2 + z +- z- + 10 5 x 1 + 2- z + + z-- 10 5- x 1- 2 + z +- z- + 10 5- x 1- 2- z + + z-- 10 5 z + z- , here we let x 2- 10 = z +- z- . Problem 2: BT, exercise 1.8 Solution: There is more than one possible formulation. Here we provide one that minimizes the total power of all the lamps with the constraints that the desired illumination of each road segment is satisfied. minimize m X j =1 p j subject to m X j =1 a ij p j I ? i , i = 1 , , n p j , j = 1 , , m. Problem 3: BT, exercise 1.9 Solution: The decision variable x ijg stands for the number of students from neighborhood i 1 going to the g-th grade of school j . The following linear programming assignes all students to schools, while minimizing the total distance traveled by all students....
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hw1_sol-1 - MIT 2.098/6.255/15.093 Optimization Methods...

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