Lecture9

# Lecture9 - C&O 355 Lecture 9 N Harvey...

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Lecture 9 N. Harvey http://www.math.uwaterloo.ca/~harvey/

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Outline Complementary Slackness “Crash Course” in Computational Complexity Ellipsoids
Duality: Geometric View We can “generate” a new constraint aligned with c by taking a conic combination (non-negative linear combination) of constraints tight at x . What if we use constraints not tight at x ? x 1 x 2 x 1 + 6x 2 · 15 Objective Function c x -x 1 +x 2 · 1

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Duality: Geometric View We can “generate” a new constraint aligned with c by taking a conic combination (non-negative linear combination) of constraints tight at x . What if we use constraints not tight at x ? x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c x Doesn’t prove x is optimal!
Duality: Geometric View What if we use constraints not tight at x ? This linear combination is a feasible dual solution, but not an optimal dual solution Complementary Slackness: To get an optimal dual solution, must only use constraints tight at x . x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c x Doesn’t prove x is optimal!

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Weak Duality Primal LP Dual LP Theorem: “Weak Duality Theorem” If x feasible for Primal and y feasible for Dual then c T x · b T y. Proof: c T x = (A T y) T x = y T A x · y T b. ¥ Since y ¸ 0 and Ax · b
Primal LP Dual LP Corollary: If x and y both feasible and c T x=b T y then x and y are both optimal. Theorem:

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Lecture9 - C&O 355 Lecture 9 N Harvey...

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