L4_func - Lecture 4: Operations Preserving Convexity Convex...

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Unformatted text preview: Lecture 4: Operations Preserving Convexity Convex Functions January 29, 2007 Lecture 4 Outline I Recession Cone Review I Lineality Space I Convex Set Decomposition I More Operations Preserving Convexity I Convex Functions I Examples I Verifying Convexity of a Function Convex Optimization 1 Lecture 4 Recession Cone of a Closed Convex Set Let X be a closed convex nonempty set in R n I A vector y is a recession direction of X if and only if there exists some vector x such that the ray { x + αy | α > } lies in the set X I The recession cone R X is closed and convex I Examples: X = { x | a x ≤ b } X = { x | x Px + a x + b ≤ } with P > ?, P ≥ ? I For two closed convex sets X 1 and X 2 with nonempty intersection X 1 ∩ X 2 , we have R X 1 ∩ X 2 = R X 1 ∩ R X 2 Th. A closed convex set is bounded if and only if R X = { } Convex Optimization 2 Lecture 4 Lineality Space Let X be a closed convex nonempty set in R n Def. The lineality space of X is the set of vectors y such that both y and- y are recession directions of X , denoted by L X . I Informally, L X is the set of directions y along which the set X is “flat” I Formally, L X = R X ∩ (- R X ) I The lineality space of X is a subspace of R n (hence, ∈ L X always) I Examples: X 1 = { x | Ax + b ≤ } X 2 = { x | x Px + a x + b ≤ } with P ≥ I For any two closed convex sets X 1 and X 2 with nonempty intersection ( X 1 ∩ X 2 6 = ∅ ), we have L X 1 ∩ X 2 = L X 1 ∩ L X 2 Convex Optimization 3 Lecture 4 Decomposition of a Convex Set Th. For any nonempty closed convex set X , we have X = L X + ( X ∩ L ⊥...
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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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L4_func - Lecture 4: Operations Preserving Convexity Convex...

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