Transcription of Lecture 2: Convex sets
{{id}} {{{paragraph}}}
Example: marketing
Lecture 2: Convex sets
Lecture 2: Convex setsAugust 28, 2008 Lecture 2 Outline Review basic topology inRn open Set and Interior Closed Set and Closure Dual Cone Convex set Cones Affine sets Half-Spaces, Hyperplanes, Polyhedra Ellipsoids and Norm Cones Convex , Conical, and Affine Hulls Simplex Verifying ConvexityConvex Optimization1 Lecture 2 Topology ReviewLet{xk}be a sequence of vectors sequence{xk} Rnconvergesto a vector x Rnwhen xk x tends to 0 ask Notation: When{xk}converges to a vector x, we writexk x The sequence{xk}converges x Rnif and only if for each componenti: thei-th components ofxkconverge to thei-th component of x|xik xi|tends to 0 ask Convex Optimization2 Lecture 2 open Set and InteriorLetX Rnbe a nonempty setXisopenif for everyx Xthere is an open ballB(x, r)thatentirely lies in the setX, ,for eachx Xthere isr > for allzwith z x < r,we havez vectorx0is aninterior pointof the setX, if there is a ballB(x0, r)contained entirely in the the setXis the set of
Lecture 2 Open Set and Interior Let X ⊆ Rn 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 > 0 s.th. for all z with kz − xk < r, we have z ∈ X Def. A vector x0 is an interior point of the set X, if there is a ball B(x0,r) contained entirely in the set X Def. The interior of the ...
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: