# L12 - CO350 Linear Programming Chapter 5 Basic Solutions...

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CO350 Linear Programming Chapter 5: Basic Solutions 30th May 2005

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Chapter 5: Basic Solutions 1 Recap Last week, we learn Deﬁnition of a basis B of matrix A . Deﬁnition of basic solution x * of Ax = b determined by basis B . x * is a basic solution if and only if { A j : x * j 6 = 0 } is linearly independent. Deﬁnition of basic feasible solution of { Ax = b, x 0 } . Deﬁnition of convex sets. Deﬁnition of extreme points of convex sets.
2 Relating bfs and extreme points Theorem 5.3 (Pg 63) Let A be m by n with rank m . Let F be the set { x : Ax = b, x 0 } . x * is a bfs of { Ax = b, x 0 } ⇐⇒ x * is an extreme point of F . Proof : “ = ” [Contradiction] Suppose x * is a bfs. x * is a feasible solution = x * F . x * is basic solution = x * is determined by some basis B . Suppose x * is not an extreme point of F . x * F = x * lie strictly between two vectors in F . I.e., there are x 1 ,x 2 F with x 1 6 = x 2 and 0 < λ < 1 such that x * = λx 1 + (1 - λ ) x 2 . For each j / B , x * j = 0 . So 0 = x * j = λx 1 j + (1 - λ ) x 2 j x 1 ,x 2 0 and 0 < λ < 1 = λx 1 j and (1 - λ ) x 2 j 0 . So λx 1 j = (1 - λ ) x 2 j = 0 . Hence x 1 j = x 2 j = 0 . Now x 1 j = x 2 j = 0 for all j / B , and also Ax 1 = Ax 2 = b . This means that x 1 and x 2 are both basic solutions of Ax = b determined by B . This contradicts

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L12 - CO350 Linear Programming Chapter 5 Basic Solutions...

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