convex functions

Convex functions - LECTURE 2 LECTURE OUTLINE Convex sets and Functions Epigraphs Closed convex functions Recognizing convex functions SOME MATH

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LECTURE 2 LECTURE OUTLINE Convex sets and Functions Epigraphs Closed convex functions Recognizing convex functions
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SOME MATH CONVENTIONS All of our work is done in < n : space of n -tuples x =( x 1 ,...,x n ) All vectors are assumed column vectors 0 ” denotes transpose, so we use x 0 to denote a row vector x 0 y is the inner product n i =1 x i y i of vectors x and y •k x k = x 0 x is the (Euclidean) norm of x .W e use this norm almost exclusively See Section 1.1 of the book for an overview of the linear algebra and real analysis background that we will use
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CONVEX SETS Convex Sets Nonconvex Sets x y α x + (1 - α )y, 0 < α < 1 x x y y x y A subset C of < n is called convex if αx +(1 α ) y C, x, y α [0 , 1] Operations that preserve convexity Intersection, scalar multiplication, vector sum, closure, interior, linear transformations Cones: Sets C such that λx C for all λ> 0 and x C (not always convex or closed)
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CONVEX FUNCTIONS α f(x) + (1 - α )f(y) xy C
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This note was uploaded on 06/16/2010 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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Convex functions - LECTURE 2 LECTURE OUTLINE Convex sets and Functions Epigraphs Closed convex functions Recognizing convex functions SOME MATH

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