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Convex Sets (chapter 2 of Convex programming)

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.

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Convex Sets (chapter 2 of Convex programming)

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  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

  5. C So C is an affine set.

  6. Convex Sets

  7. Convex combination and convex hull

  8. Convex cone

  9. Some important examples

  10. Hyperplanes and halfspaces • Open halfspace: interior of halfspace

  11. Euclidean ball and ellipsoid

  12. Norm balls and norm cones

  13. Norm balls and norm cones

  14. Polyhedra

  15. Positive semidefinite cone

  16. Operations that preserve convexity

  17. Intersection

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

  19. Affine functions

  20. Affine functions

  21. Affine functions

  22. Perspective and linear-fractional function

  23. Perspective and linear-fractional function

  24. Generalized inequalities

  25. Generalized inequalities

  26. 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].

  27. Minimum and minimal elements

  28. Separating Hyperplane theorem

  29. Separating Hyperplane theorem Here we consider a special case,

  30. Support Hyperplane theorem

  31. Dual cones and generalized inequalities

  32. Minimum and minimal elements via dual inequalities

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