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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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?...
View Full Document

Page1 / 5

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online