Practice Prelim 4 Solutions

# Practice Prelim 4 Solutions - ORIE 3310/5310 Prelim 4...

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Unformatted text preview: ORIE 3310/5310: Prelim 4 Solutions and grading scheme (problems 2, 3) 2(a) Give the cut developed by Gomory for the standard form integer programming problem with A,b,c integer-valued: max { cs | Ax = b, x ≥ ,x integral } Solution. Consider an optimal basic solution of the LP-relaxation (where the solution is not integer- valued). Let x B i be a basic variable in row i of the simplex tableau, with x B i = ¯ b i , not an integer. The row i equation has the following form: x B i + X x j nonbasic ¯ a ij x j = ¯ b i . (1) For all feasible solution x to the LP-relaxation, the following holds: x B i + X x j nonbasic b ¯ a ij c x j ≤ ¯ b i . The following may not be satisfied by all solution x to the LP relaxation, but is satisfied by all integral-valued solution x to the LP relaxation: x B i + X x j nonbasic b ¯ a ij c x j ≤ ¯ b i . (2) By equations (1) and (2), we have: X x j nonbasic (¯ a ij- b ¯ a ij c ) x j ≥ ¯ b i- ¯ b i . (3) [Total: 10 points. Seven points for correct cut/constraint as given in (3) above; 3 points for some explanation of the cut. If constraint (3) is not explicitly written but is explained well, then partial credit will be given, proportional to how clear the explanation is. An explanation is clear when it can very easily be seen that it describes inequality (3).] 2(b) Show that the slack variable for the cut from part (a) must also be integer-valued for any integral, feasible solution to the original problem. Why is this important?...
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## This note was uploaded on 10/29/2011 for the course ORIE 3310 taught by Professor Bland during the Spring '08 term at Cornell.

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Practice Prelim 4 Solutions - ORIE 3310/5310 Prelim 4...

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