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MIT6_079F09_lec02

# MIT6_079F09_lec02 - Convex Optimization Boyd Vandenberghe 2...

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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

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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 = 0 . 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 0 θ 1 convex set : contains line segment between any two points in the set x 1 , x 2 C, 0 θ 1 = θx 1 + (1 θ ) x 2 C examples (one convex, two nonconvex sets) Convex sets 2–3

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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 0 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 0 , θ 2 0 0 x 1 x 2 convex cone : set that contains all conic combinations of points in the set Convex sets 2–5

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Hyperplanes and halfspaces hyperplane : set of the form { x | a T x = b } ( a = 0 ) a x 0 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 0 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

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Norm balls and norm cones norm: a function � · � that
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MIT6_079F09_lec02 - Convex Optimization Boyd Vandenberghe 2...

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