mthsc810-lecture12-1x2

mthsc810-lecture12-1x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 12 Pietro Belotti Dept. of Mathematical Sciences Clemson University October 11, 2011 Reading for today: Sections 4.3-4.4 Reading for Oct. 13: Sections 4.5-4.6 Midterm 2: November 1, 2011, 2pm Dual of a problem in standard form If the primal is min c x s . t . A x = b x then, by applying the magic table, the dual is max u b s . t . A u c Not in standard form: apart from maximization, Dual variables are unrestricted in sign constraints (due to non-negative primal variables) Weak duality Weak duality : Given a primal min { c x : A x b , x } and its dual max { u b : A u c , u } , u b c x for any x and u feasible for their respective problems. Proof: c x ( A u ) x = u A x u b u b is a LB, our main purpose when constructing the dual. Strong duality Strong duality : If a problem min { c x : A x b , x } is bounded and its dual max { u b : A u c , u } is bounded, their optimal solutions x and u coincide in value: u b = c x Proof of strong duality Assume problem in standard form min { c x : A x = b , x } feasible and bounded....
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mthsc810-lecture12-1x2 - MthSc 810: Mathematical...

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