W 20 CSE 203 B Convex Set Xinyuan

W 20 CSE 203 B: Convex Set Xinyuan Wang 01/17/2020 1

Contents • Affine and convex sets • Example of convex sets • Key properties of convex sets • Proper cone, dual cone and generalized inequality 2

Contents • Affine and convex sets • Definitions • Affine/convex/conic combinations • Implicit expression and explicit enumeration 3

Affine set 4

Convex set 5

Convex combination and convex hull Explicit expression of convex hull The convex hull is the smallest convex set that contains 6

Cone Conic hull of sets 7

Set Specification via Implicit or Explicit Enumeration Implicit Expression Explicit Enumeration 8

Contents • Example of convex sets • Simple examples • Hyperplanes and halfspaces • Prove the convexity of sets with operations that preserve the convexity 9

Examples of convex sets Exercises: 1) Prove the convexity of the sets using the definition. 2) The above convex sets are described with implicit expression, find the explicit enumeration. 10

Hyperplane and Halfspaces 11

Hyperplane and Halfspaces • 12

Prove the convexity of polyhedron 13

Positive semidefinite cone 14

Intersections preserve convexity • 15

Contents • Key properties of convex sets • Separating hyperplane theorem • Supporting hyperplane theorem 16

Separating hyperplane theorem 17

Supporting hyperplane theorem 18

Contents • Proper cone and generalized inequality • Dual cone and generalized inequality 19

Generalized inequality Review the properties of proper cones (chap 2. 4. 1) and generalized inequality (exercise 2. 30). 20

Generalized inequality Nonnegative orthant Positive semidefinite cone 21

Minimum and minimal element 22

Dual cones 23

Dual characterization of minimum and minimal 24

reference [1] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, http: //stanford. edu/ boyd/cvxbook/, 2004. 25