This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ⊥...
View
Full
Document
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

Click to edit the document details