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.).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.3: Systems Of Inequalities

Problem 33E

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