convex sets

# convex sets - Convex Optimization — Boyd& Vandenberghe...

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Unformatted text preview: Convex Optimization — Boyd & Vandenberghe 2. Convex sets • affine and convex sets • some important examples • operations that preserve convexity • generalized inequalities • separating and supporting hyperplanes • dual cones and generalized inequalities 2–1 Affine set line through x 1 , x 2 : all points x = θx 1 + (1- θ ) x 2 ( θ ∈ R ) x 1 x 2 θ = 1 . 2 θ = 1 θ = 0 . 6 θ = 0 θ =- . 2 affine set : contains the line through any two distinct points in the set example : solution set of linear equations { x | Ax = b } (conversely, every affine set can be expressed as solution set of system of linear equations) Convex sets 2–2 Convex set line segment between x 1 and x 2 : all points x = θx 1 + (1- θ ) x 2 with ≤ θ ≤ 1 convex set : contains line segment between any two points in the set x 1 , x 2 ∈ C, ≤ θ ≤ 1 = ⇒ θx 1 + (1- θ ) x 2 ∈ C examples (one convex, two nonconvex sets) Convex sets 2–3 Convex combination and convex hull convex combination of x 1 ,. . . , x k : any point x of the form x = θ 1 x 1 + θ 2 x 2 + ··· + θ k x k with θ 1 + ··· + θ k = 1 , θ i ≥ convex hull conv S : set of all convex combinations of points in S Convex sets 2–4 Convex cone conic (nonnegative) combination of x 1 and x 2 : any point of the form x = θ 1 x 1 + θ 2 x 2 with θ 1 ≥ , θ 2 ≥ x 1 x 2 convex cone : set that contains all conic combinations of points in the set Convex sets 2–5 Hyperplanes and halfspaces hyperplane : set of the form { x | a T x = b } ( a 6 = 0 ) a x a T x = b x halfspace: set of the form { x | a T x ≤ b } ( a 6 = 0 ) a a T x ≥ b a T x ≤ b x • a is the normal vector • hyperplanes are affine and convex; halfspaces are convex Convex sets 2–6 Euclidean balls and ellipsoids (Euclidean) ball with center x c and radius r : B ( x c , r ) = { x | k x- x c k 2 ≤ r } = { x c + ru | k u k 2 ≤ 1 } ellipsoid: set of the form...
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## This note was uploaded on 06/16/2010 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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convex sets - Convex Optimization — Boyd& Vandenberghe...

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