Lecture6 - C&O 355 Mathematical Programming Fall 2010...

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Mathematical Programming Fall 2010 Lecture 6 N. Harvey
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Polyhedra Definition: For any a 2 R n , b 2 R , define Def: Intersection of finitely many halfspaces is a polyhedron Def: A bounded polyhedron is a polytope , i.e., P µ { x : -M · x i · M 8 i } for some scalar M So the feasible region of LP is polyhedron Hyperplane Halfspaces
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Convex Sets Def: Let x 1 ,…,x k 2 R n . Let ® 1 ,…, ® k satisfy ® i ¸ 0 for all i and i ® i = 1. The point i ® i x i is a convex combination of the x i ’s. Def: A set C µ R n is convex if for every x, y 2 C and 8 ® 2 [0,1], the convex combination ® x+(1- ® )y is in C. Claim 1 : Any halfspace is convex. Claim 2 : The intersection of any number of convex sets is convex. Corollary : Every polyhedron is convex. Not convex x y Convex x y
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Convex Functions Let C µ R n be convex. Def: f : C ! R is a convex function if f( ® x+(1- ® )y) · ® f(x) + (1- ® )f(y) 8 x,y 2 C Claim: Let f : C ! R be a convex function, and let a 2 R . Then { x 2 C : f(x) · a } (the “sub-level set”) is convex. Example: Let . Then f is convex. Corollary: Let B = { x : k x k · 1 }. (The Euclidean Ball) Then B is convex.
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Affine Maps Def: A map f : R n ! R m is called an affine map if f(x) = Ax + b for some matrix A and vector b. Fact:
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Lecture6 - C&O 355 Mathematical Programming Fall 2010...

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