Lecture8

Lecture8 - C&O 355 Lecture 8 N Harvey...

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C&O 355 Lecture 8 N. Harvey http://www.math.uwaterloo.ca/~harvey/

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Outline Solvability of Linear Equalities & Inequalities Farkas ’ Lemma Fourier-Motzkin Elimination
Strong Duality (for inequality form LP) 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. Weak Duality implies c T x · b T y. So strong duality says c T x ¸ b T Restatement of Theorem: Primal has an optimal solution , Dual has an optimal solution , the following system is solvable: “Solving an LP is equivalent to solving a system of inequalities” Can we characterize when systems of inequalities are solvable? (for feasible x,y) (for optimal x,y)

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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 Proof: Simple consequence of Gaussian elimination working. Perform row eliminations on augmented matrix [ A | b ], so that A becomes upper-triangular If resulting system has i th row of A equal to zero but b i non-zero then no solution exists This can be expressed as y T A=0 and y T b<0. Otherwise, back-substitution yields a solution. ¥ Or y T b>0, by negating y
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 1 x 2 span(A 1 ,…,A n ) = column space of A x 3 A 1 A 2 b
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 b y Hyperplane Positive open halfspace ++ H a,0 col-space(A) μ H y,0 but b 2 H

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Systems of Ineq ualities Lemma:
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Lecture8 - C&O 355 Lecture 8 N Harvey...

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