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mthsc810-lecture15-1x2

mthsc810-lecture15-1x2 - MthSc 810 Mathematical Programming...

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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 ⇒ ∀ λ 0 λ x C Examples: R n , R n + { ( x 1 , x 2 ) R 2 : x 1 x 2 = 0 } (non-convex) { x R n : A x 0 } for any A R m × n (is this convex?) Note: if a cone C is non-empty, 0 C .
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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 ⇒ ∀ λ 0 x + λ d P ) } The directions d of a recession cone of a polyhedron P = { x R n : A x b } must satisfy: x P ( λ 0 A ( x + λ d ) b ) Since x P , A x b λ 0 , A x b + λ A d 0 if and only if A d 0 The recession cone is { d R n : A d 0 } What is the recession cone of a bounded polyhedron?
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