# ps4 - 5. For S = { x ∈ B n | ∑ j ∈ N a j x j ≤ b }...

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Homework 4 IE418 – Integer Programming Dr. Ralphs Due April 5, 2007 1. Given a linear inequality in 0-1 variables and the set S = x B | N 1 | + | N 2 | | X j N 1 a j x j - X j N 2 a j x j b , where a j > 0 for j N 1 N 2 , write necessary and suﬃcient conditions for ˆ S = . ˆ S = B | N 1 | + | N 2 | . ˆ x j = 0 for all x S . ˆ x j = 1 for all x S . ˆ x i + x j 1 for all x S . ˆ x i x j for all x S . ˆ x i + x j 1 for all x S . 2. Show that the convex hull of feasible solutions to the integer program max { x 1 - 2 x 2 | 1 x 1 2 x 2 , x Z 2 } (1) is not a polyhedron (and that the rationality assumption is therefore necessary in order to show that the convex hull of feasible solutions to an integer program is a polyhedron). 3. Show that the C-G rank of conv( S t ) is t , where S t = P t Z 2 and P t = { x R 2 + | tx 1 + x 2 1 + t, - tx 1 + x 2 1 , x 1 1 } . 4. Find the convex hull of S = { x B 4 | x 1 + 2 x 2 + 3 x 3 + 4 x 4 4 } .
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Unformatted text preview: 5. For S = { x ∈ B n | ∑ j ∈ N a j x j ≤ b } , show that for j ∈ N , the inequalities x j ≥ 0 and x j ≤ 1 are facet-inducing for conv( S ) when a ∈ Z n + and a j + a k ≤ b for k ∈ N \ { j } . 6. The ﬁnal question is computational. You will be experimenting with the parameters of the SYMPHONY MILP solver. Following the instructions in the IE418 Wiki at https://coral.ie.lehigh.edu/projects/ie418/wiki/TuningParameters study the eﬀect of the various parameters listed in the ﬁle options.txt . Try to get the test set to solve as quickly as possible by changing only the listed parameters. 1...
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## This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .

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