Consider a set of 500 equations in 100 variables: Ax\le b, given by a_{i}^{T}x\le b_{i}. Suppose that these are not feasible, i.e., there is no solution that satisfies all of them. Show that in fact, there must be a subset of at most 101 which are not feasible. (Hint: Use duality and complementary slackness, E.g., what is the LP that finds the minimum value of slack s we have to put into Ax−1s≤b to make it feasible? The dual of such an LP could then be useful, and complementary slackness can relate variables and constraints between the two.).
Consider a set of 500 equations in 100 variables: Ax\le b, given by a_{i}^{T}x\le b_{i}. Suppose that these are not feasible, i.e., there is no solution that satisfies all of them. Show that in fact, there must be a subset of at most 101 which are not feasible. (Hint: Use duality and complementary slackness, E.g., what is the LP that finds the minimum value of slack s we have to put into Ax−1s≤b to make it feasible? The dual of such an LP could then be useful, and complementary slackness can relate variables and constraints between the two.).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Consider a set of 500 equations in 100 variables:
Ax\le b,
given by
a_{i}^{T}x\le b_{i}.
Suppose that these are not feasible, i.e., there is no solution that satisfies all of them. Show that in fact, there must be a subset of at most 101 which are not feasible. (Hint: Use duality and complementary slackness, E.g., what is the LP that finds the minimum value of slack s we have to put into Ax−1s≤b to make it feasible? The dual of such an LP could then be useful, and complementary slackness can relate variables and constraints between the two.).
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