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Unformatted text preview: Convex Optimization Boyd & Vandenberghe 2. Convex sets ane and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual cones and generalized inequalities 21 Ane set line through x 1 , x 2 : all points x = x 1 + (1 ) x 2 ( R ) = 1 . 2 x 1 = 1 = 0 . 6 x 2 = 0 = . 2 ane set : contains the line through any two distinct points in the set example : solution set of linear equations { x  Ax = b } (conversely, every ane set can be expressed as solution set of system of linear equations) Convex sets 22 Convex set line segment between x 1 and x 2 : all points x = x 1 + (1 ) x 2 with 1 convex set : contains line segment between any two points in the set x 1 , x 2 C, 1 = x 1 + (1 ) x 2 C examples (one convex, two nonconvex sets) Convex sets 23 Convex combination and convex hull convex combination of x 1 ,. . . , x k : any point x of the form x = 1 x 1 + 2 x 2 + + k x k with 1 + + k = 1 , i convex hull conv S : set of all convex combinations of points in S Convex sets 24 Convex cone conic (nonnegative) combination of x 1 and x 2 : any point of the form x = 1 x 1 + 2 x 2 with 1 , 2 x 1 x 2 convex cone : set that contains all conic combinations of points in the set Convex sets 25 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 ane and convex; halfspaces are convex Convex sets 26 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 27 Norm balls and norm cones...
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 Fall '06
 boyde

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