hw3 - IE521 Advanced Optimization Fall 2011, HW3 Due...

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IE521 Advanced Optimization Fall 2011, HW3 Due December 14 1. (20 points) Write the g ( p ) function for the following linear programs (function g ( p ) is the optimal solution to the LP in which the constraints are relaxed by paying a price p ). Simplify the function as much as possible. (a) max 2 x 1 + x 2 s.t. - x 1 + x 2 1 x 1 + x 2 3 x 1 - 2 x 2 4 x 1 ,x 2 0 (b) min y 1 - y 2 s.t. 2 y 1 + y 2 4 y 1 + y 2 1 y 1 + 2 y 2 3 y 1 ,y 2 0 2. (40 points) Find the dual problems associated with the following primal linear programs. (a) LP from the part (a) of problem 1. (b) LP from the part (b) of problem 1. (c) max 4 x 1 - x 2 + 2 x 3 s.t. x 1 + x 2 5 2 x 1 + x 2 7 2 x 2 + x 3 6 x 1 + x 3 = 4 x 2 3 are free 1 0 (d) min 4 x 1 + 2 x 2 - x 3 s.t. x 1 + 2 x 2 6 x 1 - x 2 + 2 x 3 = 8 x 3 is free 1 2 0 1
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3. (20 points) In order to show the equivalence of two equivalent problems‘ duals: (a) Find the dual of the following LP. max x 1 + 2 x 2 s.t. 3 x 1 + x 2 6 2 x 1 + x 2 = 5 x 1 ,x 2 0 (b) Find the dual of the following LP which is a reformulation of the LP in part (a).
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hw3 - IE521 Advanced Optimization Fall 2011, HW3 Due...

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