Homework 4

# Homework 4 - Math 171A: Mathematical Programming...

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Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #4 Due Monday February 9, 2009 Starred exercises require the use of Matlab . Exercise 4.1. Let F denote the feasible region F = { x R n : Ax = b } , where A be an m × n matrix of rank r and b is an m vector. Assume that F is not empty. Let A r denote an r × n matrix of r linearly independent rows of A and let b r denote the corresponding r -vector of right-hand sides. Show that F is identical to the set F r 4 = { x R n : A r x = b r } . Exercise 4.2. Suppose that the constant vector c is such that c T p 0 for all p such that Ap = 0. Show that this implies that c T p = 0 for all p such that Ap = 0. Exercise 4.3. Consider the equality-constrained linear program: minimize c T x subject to Ax = b , where A = ± - 1 5 0 1 1 3 - 1 1 4 2 ² , b = ± 4 - 5 ² and c = - 8 - 2 6 3 5 . (a)

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## Homework 4 - Math 171A: Mathematical Programming...

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