# hwk4 - Hard copy to be submitted in class on the due date...

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CS 525 - Fall 2011 - Homework 4 * assigned 10/5/11 - due 10/12/11 1. Do Exercise 3-4-2. 2. Do Exercise 3-4-3. 3. Solve the problem in Exercise 4-2-2 by adding dual labels to the tableau and applying Phase I and Phase II in the usual way. (Hint: If you need to add a row and column for Phase I, just use the usual addrow and addcol commands; the dual labels for the row and column will be left blank, which is OK.) 4. Do Exercise 4-4-3. 5. Consider the standard form LP minimize p T x subject to Ax b x 0 . (1) Let u R m , u 0. (a) Prove that if x is feasible for the LP, then it also satisﬁes the inequality u T Ax u T b . (b) Prove that for any u 0, the optimal value of the LP minimize x p T x subject to ( A T u ) T x b T u x 0 . (2) is less than or equal to the optimal value of (1).

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Unformatted text preview: * Hard copy to be submitted in class on the due date. No late homework accepted. (c) Show that (2) is bounded below if A T u ≤ p . (d) EXTRA CREDIT: Derive a necessary condition on u such that (2) is bounded below. (e) EXTRA CREDIT: When the LP is bounded, derive an expres-sion for the optimal value of (2). Your expression will depend on the vector u . (f) EXTRA CREDIT: Formulate the problem of ﬁnding the best such bound, by maximizing the lower bound over u ≥ 0 subject to the conditions when the LP (2) is bounded. (g) EXTRA CREDIT: How does the optimal value of the resulting optimization in part (f) problem compare to the optimal value of LP (1)? 2...
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hwk4 - Hard copy to be submitted in class on the due date...

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