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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 ) θ = 1 . 2 x 1 θ = 1 θ = 0 . 6 x 2 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 = 0 ) a x x a T x = b halfspace: set of the form { x  a T x ≤ b } ( a = 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  x − x c 2 ≤ r } = { x c + ru  u 2 ≤ 1 } ellipsoid: set of the form { x  ( x − x c ) T P − 1 ( x − x c ) ≤ 1 } with P ∈ S n ( i.e. , P symmetric positive definite) ++ x c other representation: { x c + Au  u 2 ≤ 1 } with A square and nonsingular Convex sets 2–7 Norm balls and norm cones...
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 Fall '06
 boyde
 Vector Space, Convex Optimization, convex sets

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