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Homework 8

# Homework 8 - (b Is the vertex optimal Explain why or why...

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Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #8 Due Friday March 6, 2009 The second midterm exam will be held in class on Wednesday, March 4. Starred exercises require the use of Matlab . Exercise 8.1. * Consider the following linear program with mixed constraints minimize x R n c T x subject to Ax = b Dx f. (8.1) (a) Consider the Portfolio Problem described on pages 32–35 of the Class Notes. Formulate the Portfolio Problem in the form ( 8.1 ). (b) Starting at the vertex x 0 = (0, 500, - 10) T , execute the steps of a “mixed-constraint” version of the simplex method that will treat problems with mixed constraints. (Do not write the equality constraint as two inequalities!) Exercise 8.2. Consider the standard-form problem of minimizing c T x subject to Ax = b , x 0, with A = 1 1 1 1 1 - 1 - 3 2 , b = 1 - 2 and c = - 1 0 - 2 0 . (a) Use any method of your choice to find a vertex for this constraint set.
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Unformatted text preview: (b) Is the vertex optimal? Explain why or why not. Exercise 8.3. * Consider the linear program minimize x 1 ,x 2 ,x 3 ,x 4-x 2-x 3-2 x 4 subject to x 2 + 2 x 3 ≤ 1 x 1 + x 2 + x 3 + x 4 ≤ 1 x 1-x 2-5 x 3 + 3 x 4 ≤ 1 x 1 ≥ x 2 ≥ x 3 ≥ x 4 ≥ . (a) Convert this problem into standard-form min c T x subject to Ax = b , x ≥ 0. (b) Compute one iteration of the standard-form simplex method for this problem, starting at the basic solution x B = ( 1 2 , 1 2 , 2) T deﬁned by columns 3, 4 and 7 of A . Show your work. Be sure to write down the objective function, basic set, nonbasic set, π-vector and reduced costs at both the beginning and end of the iteration. Check that the new iterate is feasible, with improved objective value....
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