Convex Sets(chapter 2 of Convex programming)
Keyur Desai
Advanced Machine Learning Seminar
Michigan State University
Why understand convex sets?
Outline
• Affine sets and convex sets
• Convex hull and convex cone
• Hyperplane, halfspace, ball, polyhedra etc.
• Operations that preserve convexity
• Establishing convexity
• Generalized inequalities
• Minimum and Minimal
• Separating and Supporting hyperplanes
• Dual cones and minimum-minimal
C
So C is an affine set.
Convex combination and convex hull
Some important examples
Hyperplanes and halfspaces
• Open halfspace: interior of halfspace
Euclidean ball and ellipsoid
Norm balls and norm cones
Norm balls and norm cones
Positive semidefinite cone
Operations that preserve convexity
IntersectionThm: The positive semidefinite cone is convex.
Q: Is polyhedra convex?
Q: What property does S have?A: S is closed convex.
Perspective and linear-fractional function
Perspective and linear-fractional function
Generalized inequalities
Generalized inequalities
Generalized inequalities: Example 2.16
It can be shown that K is a proper cone; its interior is the setof coefficients of polynomials that are positive on the interval [0; 1].
Minimum and minimal elements
Separating Hyperplane theorem
Separating Hyperplane theorem
Here we consider a special case,
Support Hyperplane theorem
Dual cones and generalized inequalities
Minimum and minimal elements via dual inequalities