handout9 - Math 171A Linear Programming Recap LP with...

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Math 171A: Linear Programming Lecture 9 Optimality Conditions for LP with Equality Constraints Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Monday, January 24, 2011 Recap: LP with equality constraints Linear programming with equality constraints: ELP minimize x R n c T x subject to Ax = b UCSD Center for Computational Mathematics Slide 2/30, Monday, January 24, 2011 Result x * is a minimizer of ELP if and only if c T p 0 for all p such that Ap = 0 Result x * is a minimizer of ELP if and only if there is no vector p such that c T p < 0 and Ap = 0 UCSD Center for Computational Mathematics Slide 3/30, Monday, January 24, 2011 Result c T p 0 for all p such that Ap = 0 if and only if c range( A T ) UCSD Center for Computational Mathematics Slide 4/30, Monday, January 24, 2011
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Proof: We have two statements that we must show are equivalent: (A) c T p 0 for all p such that Ap = 0 (B) c range( A T ) We must show that (A) implies (B) and (B) implies (A) In mathematical notation: (A) (B) and (B) (A) UCSD Center for Computational Mathematics Slide 5/30, Monday, January 24, 2011 First we show that (B) (A) If (B) is true, then c range( A T ). If c range( A T ) then c = A T λ for some λ R m . Then, c T p = ( A T λ ) T p = λ T ( Ap ) = λ T ( Ap ) = 0 for all p null( A ) c T p 0 for all p null( A ) (A) is true. UCSD Center for Computational Mathematics Slide 6/30, Monday, January 24, 2011 Next we show that (A) (B) . This is the same as showing that: If (B) does not hold, then (A) does not hold. We must show that (B) is not true (A) is not true In logic, this is called a contrapositive argument. UCSD Center for Computational Mathematics Slide 7/30, Monday, January 24, 2011 We must show that: (B) is not true (A) is not true (B) c range( A T ) (A) c T p 0 for all p such that Ap = 0 Taking the logical negative means that we have to show that: c 6∈ range( A T ) there exists a p such that c T p < 0 and Ap = 0 UCSD Center for Computational Mathematics Slide 8/30, Monday, January 24, 2011
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Every c R n can be written as c = c R + c N with c R range( A T ) and c N null( A ) Note that the definition of c R and c N imply that c T R c N = 0.
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