mthsc810-lecture15-2x2

mthsc810-lecture15-2x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 15 Pietro Belotti Dept. of Mathematical Sciences Clemson University October 27, 2011 Reading for today: Sections 4.8-4.10 Reading for Nov. 3: Sections 5.1, 5.2 Cones Definition A cone is a set C R n such that x C x C Examples: R n , R n + { ( x 1 , x 2 ) R 2 : x 1 x 2 = } (non-convex) { x R n : A x } for any A R m n (is this convex?) Note: if a cone C is non-empty, C . Recession cones Definition The recession cone of a polyhedron P R n is the set of direc- tions d on which we can move indefinitely: { d R n : ( x P x + d P ) } The directions d of a recession cone of a polyhedron P = { x R n : A x b } must satisfy: x P ( A ( x + d ) b ) Since x P , A x b , A x b + A d 0 if and only if A d The recession cone is { d R n : A d } What is the recession cone of a 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-lecture15-2x2 - MthSc 810: Mathematical...

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