mthsc810-lecture12-2x2

# mthsc810-lecture12-2x2 - 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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## This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.

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mthsc810-lecture12-2x2 - MthSc 810: Mathematical...

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