L3_sets_geom - Lecture 3 Convex Set Topology and Operations...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 3: Convex Set Topology and Operations Preserving Convexity January 24, 2007
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 3 Outline Review basic topology in R n Open Set and Interior Closed Set and Closure Dual Cone Recession Directions of a Convex Set Lineality Space Convex Set Decomposition Operations Preserving Convexity Affine Transformation Perspective Transformation Convex Optimization 1
Image of page 2
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 x k - ˆ x tends to 0 as k → ∞ Notation: When { x k } converges to a vector ˆ x , we write x k ˆ x The sequence { x k } converges ˆ x R n if and only if for each component i : the i -th components of x k converge to the i -th component of ˆ x | x i k - ˆ x i | tends to 0 as k → ∞ Convex Optimization 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 > 0 s.th. for all z with z - x < r, we have z X Def. A vector x 0 is an interior point of the set X , if there is a ball B ( x 0 , 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 ) How is int ( X ) related to X ? Example X = { x R 2 | x 1 0 , x 2 > 0 } int ( X ) = { x R 2 | x 1 > 0 , x 2 > 0 } int ( S ) of a probability simplex S = { x R n | x 0 , e x = 1 } Th. For a convex set X , the interior int ( X ) is also convex Convex Optimization 3
Image of page 4
Lecture 3 Closed Set Def. The complement of a given set X R n is the set of all vectors that do not belong to X : the complement of
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern