L2_sets - Lecture 2: Convex sets January 22, 2007 Lecture 2...

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Unformatted text preview: Lecture 2: Convex sets January 22, 2007 Lecture 2 Outline I Vectors and Norms I Convex set I Cones I Affine sets I Half-Spaces, Hyperlanes, Polyhedra I Ellipsoids and Norm Cones I Convex, Conical, and Affine Hulls I Simplex Convex Optimization 1 Lecture 2 Notation: Vectors and Norms We consider the space of n-dimensional real vectors denoted by R n I Vector x R n is viewed as a column: x = x 1 . . . x n I The prime denotes a transpose: x = [ x 1 ,...,x n ] I For vectors x R n and y R n , the inner product is: x y = n i =1 x i y i I Norm (length) of a vector: Euclidian norm k x k = x x = q n i =1 x 2 i ( l 2-norm) Sum-norm k x k 1 = n i =1 | x i | ( l 1-norm) Max-norm k x k = max 1 i n | x i | ( l -norm) Convex Optimization 2 Lecture 2 Norm and Distance I Properties of a norm Nonnegativity k x k for every x R n Uniqueness of zero-value k x k = 0 if and only if x = 0 Triangle relation k x k k x- y k + k y k for every x and y R n I (Euclidian) Distance between two vectors x R n and y R n k x- y k = v u u t n X i =1 ( x i- y i ) 2 I (Euclidian) Ball centered at x R n with a radius r > { x R n | k x- x k r } (closed ball) { x R n | k x- x k < r } (open ball) Convex Optimization 3 Lecture 2 Convex set I A line segment defined by vectors x and y is the set of points of the form...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.

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L2_sets - Lecture 2: Convex sets January 22, 2007 Lecture 2...

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