mthsc810-lecture17-2x2

mthsc810-lecture17-2x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 17 Pietro Belotti Dept. of Mathematical Sciences Clemson University November 10, 2011 Reading for today: Sections 5.2-5.5 November 13: Class canceled Reading for Nov. 17: Sections 7.1, 7.2, 7.5 Homework #11 is out! Due next Thursday. Value functions in Optimization Consider the function F ( b ) = min { c ⊤ x : A x = b , x ≥ } . Note: F ( α b ) = α F ( b ) for any α ≥ 0. Also, ◮ F ( b ) is a function of m variables. ◮ It is defined on S = { y ∈ R m : y = A x , x ∈ R n + } . ⇒ S is a (polyhedral and thus convex) cone. ◮ F ( b ) = + ∞ for b such that { x ∈ R n + : A x = b } = ∅ . ◮ F ( b ) = −∞ for b such that { x ∈ R n + : A x = b } contains an extreme ray d ∈ R n such that c ⊤ d < 0. ◮ Can these three cases occur for given A , c ? How do we compute F ( b ) for a given b ? Well, by solving an optimization problem. ◮ Suppose B is an optimal basis (for b ). ◮ Then, for its associated submatrix B , x ⋆ B = B − 1 b . ◮ The objective function of the optimal solution is c ⊤ B B − 1 b = ( p ⋆ ) ⊤ b . ( p ⋆ ) ⊤ = c BB − 1 is the vector of optimal dual variables....
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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-lecture17-2x2 - MthSc 810: Mathematical...

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