a6 - max c T x Ax = b,x 0(P and its dual(D Let B be a basis...

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CO350 LINEAR OPTIMIZATION - HW 6 Due Date: Friday June 22nd, at 2pm, in the drop box outside the Tutorial Center. Recall, late assignments will not be graded. Exercise 1. Consider the following linear program (P), max x 1 - x 2 subject to x 1 + x 2 3 ( a ) - x 1 + x 2 1 ( b ) x 2 2 ( c ) x 1 , x 2 0 (1) Draw the feasible region for (P). (2) Rewrite (P) in standard equality form by introducing slack variable x 3 for constraint (a), x 4 for (b) and x 5 for (c). (3) Write the tableau T corresponding to the basis B = { 1 , 2 , 3 } . What is the basic solution x * for B ? (4) Indicate in the feasible region of (P) the point corresponding to x * . (5) Introduce perturbation ²,² 2 3 to the tableau T . (6) Find an optimal solution using the simplex method starting from the perturbed tableau. Exercise 2. Consider the following linear programs.
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Unformatted text preview: max( c T x : Ax = b,x 0) (P) and its dual (D). Let B be a basis of A , let x * be a basic solution for B and let y * := ( A T B )-1 c B . (1) Show that x * and y * satisfy the complementary slackness conditions for (P) and (D). (2) Show that c N if and only if y * is feasible for (D). (3) Deduce the (Strong) Duality theorem from (1) and (2). IMPORTANT: You answers should be self contained for part (1) and part (2). Exercise 3. (1) Can the entering variable in one iteration of the simplex method be the leaving variable in the next iteration? (2) Can the leaving variable in one iteration be the entering variable in the next iteration? 1...
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