Homework 5

# Homework 5 - (4 Let B be an optimal basis for(P What is the...

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CO350 LINEAR OPTIMIZATION - HW 5 Due Date: Monday February 12th at the beginning of class. Please do NOT leave the assignments in the drop off box but return them to the instructor in class. From the notes: exercise 6.9.7 exercise 6.9.9 Exercise 1. Do exercise 6.9.10 from the notes. Find an optimal dual solution for the linear program. Exercise 2. Consider the following linear program (P) in standard equality form: max { c T x : Ax = b,x 0 } Consider a diagonal matrix, D with non-zero diagonal entries d 1 ,d 2 ,...,d m . Let A 0 := DA and b 0 := DB (i.e. the system A 0 x = b 0 is obtained from Ax = b by multiplying the ﬁrst constraint by d 1 the second by d 2 etc. ..). Denote by (P’) the linear program (P’): max { c T x : A 0 x = b 0 ,x 0 0 } (1) Show that B is a basis for A if and only if it is a basis for A 0 . (2) Consider the tableau for basis B of (P) 1 0 - ¯ c N z * 0 I ¯ A N ¯ b What is the relation between the tableau corresponding to basis B for (P) and (P’)? Justify your answer! (3) What is the relation between the optimal solutions of (P) and (P’)?
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Unformatted text preview: (4) Let B be an optimal basis for (P). What is the relation between the optimal dual solution y corre-sponding to B of (P) and the optimal dual solution y corresponding to B of (P’)? Justify your answer! Exercise 3. Let (P) be a linear program and let (D) be its dual. Let x * be a feasible solution for (P) and let y * be a feasible solution for (D). Show that if all variables of (P) and (D) are free (unrestricted) then x * is an optimal solution to (P) and y * is an optimal solution to (D). Exercise 4. Write the dual of the following linear program: max c T x + e T y s.t. Ax + By = b Cx + y ≤ d x ≥ . where A ∈ R m × n ,B ∈ R m × n ,C ∈ R n × n ,c ∈ R n ,e ∈ R n ,b ∈ R m , and d ∈ R n . 1...
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