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Transcript of Convex Sets (chapter 2 of Convex programming) Keyur Desai Advanced Machine Learning Seminar Michigan...
Convex Sets(chapter 2 of Convex programming)
Keyur Desai
Advanced Machine Learning Seminar
Michigan State University
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
IntersectionThm: The positive semidefinite cone is convex.
Q: Is polyhedra convex?
Q: What property does S have?A: S is closed convex.
Generalized inequalities: Example 2.16
It can be shown that K is a proper cone; its interior is the setof coefficients of polynomials that are positive on the interval [0; 1].