Lecture5 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 5 N. Harvey
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Review of our Theorems Fundamental Theorem of LP: Every LP is either Infeasible, Unbounded, or has an Optimal Solution. Not Yet Proven! Weak Duality Theorem: If x feasible for primal and y feasible for dual then c T x · b T y. Strong LP Duality Theorem: 9 x optimal for primal ) 9 y optimal for dual. Furthermore, c T x=b T y. Since the dual of the dual is the primal, we also get: 9 y optimal for dual ) 9 x optimal for primal. Primal LP: Dual LP:
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Variant of Strong Duality Fundamental Theorem of LP: Every LP is either Infeasible, Unbounded, or has an Optimal Solution. Variant of Strong LP Duality Theorem: If primal is feasible and dual is feasible, then 9 x optimal for primal and 9 y optimal for dual. Furthermore, c T x=b T y. Proof: Since primal and dual are both feasible, primal cannot be unbounded. (by Weak Duality) By FTLP, primal has an optimal solution x. By Strong Duality, dual has optimal solution y and c T x=b T y. ¥ Primal LP: Dual LP:
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Proof of Variant from Farkas’ Lemma Theorem: If primal is feasible and dual is feasible, then 9 x optimal for primal and 9 y optimal for dual. Furthermore, c T x=b T y. Primal LP: Dual LP: Existence of optimal solutions is equivalent to solvability of { Ax · b, A T y = c, y ¸ 0, c T x ¸ b T y } We can write this as: Suppose this is unsolvable. Farkas’ Lemma: If Mp · d has no solution, then 9 q ¸ 0 such that q T M=0 and q T d < 0.
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Farkas’ Lemma: If Mp · d has no solution, then 9 q ¸ 0 such that q T M=0 and q T d < 0. So if this is unsolvable, then there exists [ u, v
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Lecture5 - C&amp;O 355 Mathematical Programming Fall 2010...

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