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

Polyhedra

Positive semidefinite cone

Operations that preserve convexity

Intersection

Thm: The positive semidefinite cone is convex. Q: Is polyhedra convex? Q: What property does S have? A: S is closed convex.

Affine functions

Perspective and linear-fractional function

Generalized inequalities

Generalized inequalities: Example 2.16 It can be shown that K is a proper cone; its interior is the set of coefficients of polynomials that are positive on the interval [0; 1].

Minimum and minimal elements

Separating Hyperplane theorem

Here we consider a special case,

Support Hyperplane theorem

Dual cones and generalized inequalities

Minimum and minimal elements via dual inequalities