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asst5sol

asst5sol - C&O 350 Linear Optimization – Winter 2009 –...

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C&O 350 Linear Optimization – Winter 2009 – Solutions to Assignment 5 Question # Max. marks Part marks 1 4 2 16 4 7 5 3 8 4 4 4 16 6 10 5 12 6 2 2 2 Total 56 1. (Duality) 4 marks Consider the LP problem ( P ) below. max c T x Ax - y b Bx + Cy = f y 0 where A is an m × n matrix, B is an n × n matrix, and C is an n × m matrix, c, x, f are n -dimensional vectors and y, b are m -dimensional vectors. Write down the dual of ( P ). Solution : Expanding the table in page 41 of the notes, we can write the dual of ( P ) as minimize b T u + f T v A T u + B T v = c - u + C T v 0 u 0 where u is an m -dimensional vector and v is an n -dimensional vector. 2. (Complementary Slackness) 16 marks = 4+7+5 part (a): 2 marks for the dual and 2 marks for the CS conditions part (b): 1+3+3 part (c): 1+4 Consider the linear programming problem ( P ) maximize z = c T x such that Ax = b, x 0 , where A = 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 - 1 - 2 2 - 3 0 2 6 , b = 7 2 7 0 , and c T = [ - 2 , 2 , 0 , 4 , 5 , 0] . (a) Write down the dual problem ( D ) of ( P ), and then write down all the complementary slackness conditions for ( P ) and ( D ). (b) Use part (a) to determine whether or not the following solutions are optimal for ( P ). 1

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i. x 0 = [0 , 7 , 3 . 6 , 8 . 6 , - 1 . 6 , 0] T ii. x 00 = [4 . 5 , 2 . 5 , 0 , 5 , 2 , 0] T iii. x 000 = [3 . 5 , 3 . 5 , 1 , 6 . 5 , 0 . 75 , 0 . 25] T . (c) Use part (a) to determine whether or not the following solutions are optimal for ( D ). i. y 0 = [ - 12 5 , 3 5 , 4 , 1 5 ] T ii. y 00 = [0 , 3 , 4 , 1] T Solution : (a) The dual is minimize 7 y 1 + 2 y 2 + 7 y 3 subject to y 1 - 2 y 4 - 2 y 1 + 2 y 4 2 y 2 - 3 y 4 0 y 3 4 y 2 + y 3 + 2 y 4 5 y 2 - y 3 + 6 y 4 0 The complementary slackness conditions are x 1 = 0 or y 1 - 2 y 4 = - 2 x 2 = 0 or y 1 + 2 y 4 = 2 x 3 = 0 or y 2 - 3 y 4 = 0 x 4 = 0 or y 3 = 4 x 5 = 0 or y 2 + y 3 + 2 y 4 = 5 x 6 = 0 or y 2 - y 3 + 6 y 4 = 0 (b) i. x 5 < 0, so x 0 is not feasible, thus not optimal.
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asst5sol - C&O 350 Linear Optimization – Winter 2009 –...

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