Lecture4 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 4 N. Harvey
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Outline Farkas’ Lemma Fourier-Motzkin Elimination Proof of Farkas’ Lemma Proof of Strong LP Duality
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Strong Duality Primal LP: Dual LP: Strong Duality Theorem: Primal has an opt. solution x , Dual has an opt. solution y. Furthermore, optimal values are same: c T x = b T y. Our Goals: Understand when Primal and Dual have optimal solutions Compute those optimal solutions x is optimal for Primal and y is optimal for Dual , x and y are solutions to these inequalities: Can we characterize when systems of inequalities are solvable?
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Systems of Eq ualities Lemma: Exactly one of the following holds: There exists x satisfying Ax=b There exists y satisfying y T A=0 and y T b>0 Geometrically… x 1 x 2 span(A 1 ,…,A n ) = column space of A x 3 A 1 A 2 b (b is in column space of A)
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Systems of Eq ualities Lemma: Exactly one of the following holds: There exists x satisfying Ax=b (b is in column space of A) There exists y satisfying y T A=0 and y T b>0 (or it is not) Geometrically… x 2 column space of A x 3 A 1 A 2 b x 1 y Hyperplane Positive open halfspace ++ H a,0 col-space(A) µ H y,0 but b 2 H y,0
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Systems of Ineq ualities Lemma: Exactly one of the following holds: There exists x ¸ 0 satisfying Ax=b There exists y satisfying y T A ¸ 0 and y T b<0 Geometrically… Let cone(A 1 ,…,A n ) = { § i x i A i : x ¸ 0 } “cone generated by A 1 ,…,A n (Here A i is the i th column of A) x 1 x 2 x 3 A 1 A 2 b A 3 cone(A 1 ,…,A n ) (b is in cone(A 1 ,…,A n ))
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Systems of Ineq ualities Lemma: Exactly one of the following holds: There exists x ¸ 0 satisfying Ax=b (b is in cone(A 1 ,…,A n )) There exists y satisfying y T A ¸ 0 and y T b<0 Geometrically… x 1 x 2 x 3 A 1 A 2 b A 3 cone(A 1 ,…,A n ) Positive closed halfspace Negative open halfspace -- + H y,0 cone(A 1 , ,A n ) 2 H y,0 but b 2 H y,0 (y gives a separating hyperplane ) y
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Lemma: Exactly one of the following holds: There exists x
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.

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Lecture4 - C&O 355 Mathematical Programming Fall 2010...

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