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Unformatted text preview: MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 2. Solution Problem 1: BT, exercise 3.2 Solution. (a) If feasible solution x is optimal, suppose there exists a feasible direction d such that c d < 0, then for a small enough ² > 0, x + ²d is still feasible, but c ( x + ²d ) = c x + ²c d < c x contradiction. If c d ≥ 0 for every feasible direction d at x , for any point y ∈ P and y 6 = x , d = y x is a feasible direction. c y c x = c d ≥ 0, which implies x is optimal. (b) The proof is almost identical. If feasible solution x is the unique optimal solution, suppose there exists a feasible direction d such that c d ≤ 0, then for a small enough ² > 0, x + ²d is still feasible, but c ( x + ²d ) = c x + ²c d ≤ c x contradict with x being the unique optimal solution. If c d > 0 for every feasible direction d at x , for any point y ∈ P and y 6 = x , d = y x is a feasible direction. c y c x = c d > 0, which implies x is uniquely optimal. / Problem 2: BT, exercise 3.17 Solution. The initial tableau in Phase I is 5 1 1 3 1 2 x 6 = 2 1 * 3 4 1 1 x 7 = 2 1 2 3 1 1 x 8 = 1 1 4 3 1 The final tableau in Phase I is 1 7 2 1 x 1 = 2 1 3 4 1 1 x 7 = 0 1 * 7 1 1 x 3 = 1 1 / 3 1 4 / 3 1 / 3 1 / 3 1 / 3 Drive x 7 out of the basis 1 1 1 1 x 1 = 2 1 17 1 2 3 x 2 = 0 1 7 1 1 x 3 = 1 1 11 / 3 1 / 3 2 / 3 1 / 3 1 / 3 The initial tableau in Phase II is: 7 3 5 x 1 = 2 1 17 1 x 2 = 0 1 7 x 3 = 1 1 11 / 3 1 / 3 Final tableau is 3 5 82 / 7 x 5 = 2 1 17 / 7 1 x 4 = 0 1 / 7 1 x 3 = 1 / 3 1 / 3 4 / 3 1 / Problem 3: BT, exercise 3.19 Solution. (a) The current solution is optimal but the current basis is not. Thus the current solution is a degenerate optimal solution. So we have β = 0. Update the tableau using one simplex iteration 10 δ + 2 γ 3 2 / 3 x 3 = 4 1 γη 3 1 η 3 x 4 = 1 α + 4 γ 3 1 4 / 3 x 2 = 0 γ 3 1 1 / 3 There are multiple optimal solutions, thus δ + 2 γ 3 = 0. In addition, we need to have a feasible direction, which requires γ 3 ≤ 0. The conditions could be β = 0 ,δ + 2 γ 3 = 0, and γ ≤ 0. (b) The optimal cost is∞ when we have a feasible solution in the current tableau, a nonbasic variable x i with ¯ c i < 0 and u i = B 1 A i ≤ 0. We need β ≥ 0 for problem feasiblity. The variable 2 x 2 cannot satisfy all the conditions for the∞ cost. For the variable x 1 , the conditions then can be expressed as follows: α ≤ ,λ ≤ 0, and...
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This note was uploaded on 02/15/2011 for the course EECS 6.231 taught by Professor Bertsekas during the Spring '10 term at MIT.
 Spring '10
 Bertsekas
 Optimization

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