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Lecture3 - C&O 355 Mathematical Programming Fall 2010...

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Image: http://www.flickr.com/photos/singapore2010/4902039196/ C&O 355 Mathematical Programming Fall 2010 Lecture 3 N. Harvey
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C&O 355 Mathematical Programming Fall 2010 Lecture 1 N. Harvey
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Duality: Geometric View Suppose c=[-1,1] Then every feasible x satisfies c T x = -x 1 +x 2 · 1 If this constraint is tight at x ) x is optimal x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c i.e. -x 1 +x 2 =1 i.e. x lies on the red line (because equality holds here)
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Duality: Geometric View x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c Suppose c=[1,6] Then every feasible x satisfies c T x = x 1 +6x 2 · 15 If this constraint is tight at x ) x is optimal (because equality holds here) i.e. x 1 +6x 2 =15 i.e. x lies on the red line
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Duality: Geometric View x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c (because equality holds here) Suppose c= ® ¢ [1,6], where ® ¸ 0 Then every feasible x satisfies c T x = ® ¢ ( x 1 +6x 2 ) · 15 ® If this constraint is tight at x ) x is optimal i.e. x 1 +6x 2 =15 i.e. x lies on the red line
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Duality: Geometric View What if c does not align with any constraint? Can we “generate” a new constraint aligned with c? x 1 x 2 -x 1 +x 2 · 1 x 1 + 6x 2 · 15 Objective Function c
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Duality: Geometric View Can we “generate” a new constraint aligned with c? One way is to “average” the tight constraints Example: Suppose c = u+v . Then every feasible x satisfies c T x = ( u + v ) T x = (-x 1 +x 2 ) + (x 1 +6x 2 ) · 1 + 15 = 16 x is feasible and both constraints tight ) x is optimal x 1 x 2 -x 1 +x 2 · 1 x 1 +6x 2 · 15 Objective Function c u=[-1,1] v=[1,6]
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Duality: Geometric View Can we “generate” a new constraint aligned with c? One way is to “average” the tight constraints More generally: Suppose c = ® u + ¯ v for ® , ¯ ¸ 0 Then every feasible x satisfies c T x = ( ® u + ¯ v ) T x = ® (-x 1 +x 2 ) + ¯ (x 1 +6x 2 ) · ® +15 ¯ x is feasible and both constraints tight ) x is optimal x 1 x 2 -x 1 +x 2 · 1 x 1 +6x 2 · 15 Objective Function c u=[-1,1] v=[1,6]
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x feasible ) a 1 T x · b 1 and a 2 T x · b 2 ) (a 1 +a 2 ) T x · b 1 +b 2 (new valid constraint) More generally, for any ¸ 1 ,…, ¸ m ¸ 0 x feasible ) ( § i ¸ i a i ) T x · § i
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