L13 - CO350 Linear Programming Chapter 5 Basic Solutions...

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Unformatted text preview: CO350 Linear Programming Chapter 5: Basic Solutions 1st June 2005 Chapter 5: Basic Solutions 1 Recap On Monday, we learned • Theorem 5.3 Consider an LP in SEF with rank ( A ) = # rows . Then x * is bfs ⇐⇒ x * is extreme point of the feasible region . • Definition of basic feasible solution for LP problems in SIF. • Theorem 5.4 Consider an LP in SIF. Then x * is bfs ⇐⇒ x * is extreme point of the feasible region . Chapter 5: Basic Solutions 2 Why consider basic feasible solutions? Theorem 5.5 (Pg 65) Let A be m by n with rank m . Consider the LP in SEF ( P ) max. c T x s.t. Ax = b x ≥ If ( P ) has an optimal solution then ( P ) has an optimal solution that is basic. Proof : (Almost the same as the proof on page 66) Key ingredient: Show that if x * is optimal but not basic, then there is an optimal solution with more zeros entries than x * . Suppose that x * is optimal but not basic. From proof of previous theorem (also emphasized in the remarks after proof), we know that ∃ d ∈ R n , d 6 = 0 , d j = 0 whenever x * j = 0 , and Ad = 0 , and ∃ ε > , both x 1 = x * + εd and x 2 = x *- εd are feasible. We now show that c T d = 0 . If c T d > , then c T x 1 = c T x * + εc T d > c T x * contradicts x * is optimal....
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This note was uploaded on 06/11/2011 for the course C 350 taught by Professor Wolkowicz during the Fall '97 term at Waterloo.

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L13 - CO350 Linear Programming Chapter 5 Basic Solutions...

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