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
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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