Xinyuan Wang 01 17 20 20 1 Contents Affine and convex sets Example of convex sets Key properties of convex sets Proper cone dual cone and generalized ID: 1044679
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1. W20 CSE 203B:Convex SetXinyuan Wang01/17/20201
2. ContentsAffine and convex setsExample of convex setsKey properties of convex setsProper cone, dual cone and generalized inequality2
3. ContentsAffine and convex setsDefinitionsAffine/convex/conic combinationsImplicit expression and explicit enumeration3
4. Affine set4
5. Convex set5
6. Convex combination and convex hull6The convex hull is the smallest convex set that contains Explicit expression of convex hull
7. Cone7Conic hull of setsA cone formed with
8. Set Specification via Implicit or Explicit EnumerationImplicit ExpressionExplicit Enumeration 8 Implicit Expression:ConstraintsMin Subject to Explicit Expression: EnumerationMin
9. ContentsExample of convex setsSimple examplesHyperplanes and halfspacesProve the convexity of sets with operations that preserve the convexity9
10. Examples of convex sets10Exercises: Prove the convexity of the sets using the definition.The above convex sets are described with implicit expression, find the explicit enumeration.
11. Hyperplane and Halfspaces11
12. Hyperplane and Halfspaces12Example of plane in Implicit express of the hyperplane where Let be any point in the hyperplane, then theexpression becomes Q: Write the explicit expression of the hyperplane.Find the distance from a point to the plane. Normal vector
13. Prove the convexity of polyhedron13Use the definition of convex set:Proof: pick any two points in the polyhedron , for is the point ?
14. Positive semidefinite cone14
15. Intersections preserve convexity15Polyhedron: intersection of halfspaces and hyperplanes (convex) Is the intersection of a finite number of convex sets still convex? prove with the definition of convex sets.Operations that preserve convexity:IntersectionAffine functionPerspective functionsLinear-fractional functions
16. ContentsKey properties of convex setsSeparating hyperplane theoremSupporting hyperplane theorem16
17. Separating hyperplane theorem17
18. Supporting hyperplane theorem18
19. ContentsProper cone and generalized inequalityDual cone and generalized inequality19
20. Generalized inequality20 Review the properties of proper cones (chap 2.4.1) and generalized inequality (exercise 2.30).
21. 21Nonnegative orthantPositive semidefinite conea partial ordering on analogy to ordinary inequality Generalized inequality
22. Minimum and minimal element22
23. Dual cones23 Find the dual cones of Subspace Nonnegative orthant Positive semidefinite cone Review the properties of dual cones (exercise 2.31).
24. Dual characterization of minimum and minimal24 Supporting hyperplane at Supporting hyperplane at
25. reference[1] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, http://stanford.edu/ boyd/cvxbook/, 2004.25