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Unformatted text preview: Lecture 3: Convex Set Topology and Operations Preserving Convexity January 24, 2007 Lecture 3 Outline I Review basic topology in R n I Open Set and Interior I Closed Set and Closure I Dual Cone I Recession Directions of a Convex Set I Lineality Space I Convex Set Decomposition I Operations Preserving Convexity I Affine Transformation I Perspective Transformation Convex Optimization 1 Lecture 3 Topology Review Let { x k } be a sequence of vectors in R n Def. The sequence { x k } ⊆ R n converges to a vector ˆ x ∈ R n when k x k ˆ x k tends to 0 as k → ∞ I Notation: When { x k } converges to a vector ˆ x , we write x k → ˆ x I The sequence { x k } converges ˆ x ∈ R n if and only if for each component i : the ith components of x k converge to the ith component of ˆ x  x i k ˆ x i  tends to 0 as k → ∞ Convex Optimization 2 Lecture 3 Open Set and Interior Let X ⊆ R n be a nonempty set Def. The set X is open if for every x ∈ X there is an open ball B ( x,r ) that entirely lies in the set X , i.e., for each x ∈ X there is r > s.th. for all z with k z x k < r, we have z ∈ X Def. A vector x is an interior point of the set X , if there is a ball B ( x ,r ) contained entirely in the set X Def. The interior of the set X is the set of all interior points of X , denoted by int ( X ) I How is int ( X ) related to X ? I Example X = { x ∈ R 2  x 1 ≥ , x 2 > } int ( X ) = { x ∈ R 2  x 1 > , x 2 > } int ( S ) of a probability simplex S = { x ∈ R n  x , e x = 1 } Th. For a convex set X , the interior int ( X ) is also convex Convex Optimization 3 Lecture 3 Closed Set...
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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.
 Spring '07
 AngeliaNedich

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