hw3 - n-vector x ²or which Ax ≤ 0 and cx> 0(b Suppose...

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OR 3300 Optimization I Summer 2008 HW 3 LP Duality; Transportation Problem Due date: Monday, June 16, by 3:20pm in the drop box on 2nd foor Rhodes. References: Lecture notes; Bradley, Hax and Magnanti chapters 4,8. 1. BHM IV/1. 2. BHM IV/2. 3. BHM IV/4, notice typo: in the Frst constraint the coe±cient o² x 2 should be 3 instead o² 1. 4. Consider the primal LP problem, where a is an unspeciFed coe±cient: min 6 x 1 - 3 x 2 - 3 x 1 + x 2 3 4 x 1 + ax 2 4 x 1 0 , x 2 0 (a) State the dual problem. (b) ³or what values o² a is the primal problem ²easible? (c) ³or what values o² a is the dual problem ²easible? (d) ³or what values o² a does the primal problem have a Fnite optimal solution? (HINT: Pictures are very help²ul ²or this problem.) 5. Consider the system o² linear inequalities { yA = c, y 0 } , where A is an m × n matrix and c is an n -vector. (a) Suppose the linear system is ²easible , i.e., there is an m -vector y so that yA = c, y 0 . Show that there can be no
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Unformatted text preview: n-vector x ²or which Ax ≤ 0 and cx > 0 . (b) Suppose the linear system is in²easible , i.e., there is no m-vector y so that yA = c, y ≥ 0 . Show that in this case there is an n-vector x so that Ax ≤ 0 and cx > 0 . (HINT: Consider the LP { min y s.t. yA = c, y ≥ } 6. BHM IV/15. Hand in all parts except d. 7. BHM IV/17. 8. Solve the linear program min 2 x 1 + 3 x 2 + 4 x 3 x 1 + 2 x 2 + x 3 ≥ 3 + 2 θ 2 x 1-x 2 + 3 x 3 ≥ 4-θ x 1 ≥ , x 2 ≥ , x 3 ≥ ²or all values o² θ ! Plot the objective ²unction in terms o² θ . 9. BHM VIII/2, VIII/8, VIII/16. 10. BHM VIII/10. Solve via either algorithm we studied ²or the transportation model; use the minimum (maximum) matrix rule to Fnd an initial ²easible assignment. 11. BHM VIII/15, notice typo: The corresponding unit costs ²rom . .. ....
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