1. Convex Sets (chapter 2 of Convex programming) Keyur Desai Advanced Machine Learning Seminar Michigan State University

2. Why understand convex sets?

3. 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

4. Affine Sets

6. C So C is an affine set.

8. Convex Sets

9. Convex combination and convex hull

10. Convex cone

11. Some important examples

12. Hyperplanes and halfspaces Open halfspace: interior of halfspace

13. Euclidean ball and ellipsoid

14. Norm balls and norm cones

16. Polyhedra

17. Positive semidefinite cone

18. Operations that preserve convexity

19. Intersection

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

21. Affine functions

24. Perspective and linear-fractional function

26. Generalized inequalities

28. 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].

29. Minimum and minimal elements

30. Separating Hyperplane theorem

31. Here we consider a special case,

32. Support Hyperplane theorem

33. Dual cones and generalized inequalities

34. Minimum and minimal elements via dual inequalities