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

Affine Sets

C

So C is an affine set.

Convex Sets

Convex combination and convex hull

Convex cone

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

Polyhedra

Positive semidefinite cone

Operations that preserve convexity

Intersection

IntersectionThm: The positive semidefinite cone is convex.

Q: Is polyhedra convex?

Q: What property does S have?A: S is closed convex.

Affine functions

Affine functions

Affine functions

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