mthsc810-lecture17-1x2

mthsc810-lecture17-1x2 - MthSc 810 Mathematical Programming...

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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: Classcanceled 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 0 } . 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 ?
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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 B B 1 is the vector of optimal dual variables. p is a gradient : for b b + β , with small β , objective function changes by p β .
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