This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CO 355 Lecture 13 Convex functions Vris Cheung (Substitute class) October 26, 2010 Vris Cheung (University of Waterloo) CO 355 2010 1 / 22 Preliminaries Outline 1 Preliminaries 2 Convex functions 3 Equivalent conditions of convexity Vris Cheung (University of Waterloo) CO 355 2010 2 / 22 Preliminaries Basic notions (that you should know) + openness / closedness of a set + interior of a set + closure of a set + (in Euclidean space) a set is compact iff it is closed + bounded . + BolzanoWeierstrass theorem : A sequence in a compact set has a convergent subsequence. + Continuity of functions Vris Cheung (University of Waterloo) CO 355 2010 3 / 22 Preliminaries Calculus review + f : S R n R is differentiable at x int ( S ) if d R n s.t. lim x x f ( x )  f ( x )  d > ( x x ) k x x k = 0. Here d is called the gradient of f at x . Notation : f ( x ) . + Recall: f ( x ) = f x 1 ( x ) , f x 2 ( x ) , . . . , f x n ( x ) . + If the maps x 7 f x i ( x ) ( i = 1, . . . , n ) are defined in a nbd. of and are continuous at x , then f is continuously differentiable at x . Vris Cheung (University of Waterloo) CO 355 2010 4 / 22 Preliminaries Calculus review If the maps x 7 f x i ( x ) ( i = 1, . . . , n ) are differentiable at x , we may define the Hessian of f at x as the matrix 2 f ( x ) R n n by 2 f ( x ) ij := 2 f x i x j ( x ) . If f is twice continuously differentiable at x , the Hessian is symmetric. * In the course notes, the Hessian is denoted by Hf ( x ) . Vris Cheung (University of Waterloo) CO 355 2010 5 / 22 Preliminaries Calculus review Let S R n be nonempty open, x int ( S ) and f : S R be given. + Gradient of f at x : f ( x ) = f x 1 ( x ) , f x 2 ( x ) , . . . , f x n ( x ) . + Hessian of f at x : the matrix 2 f ( x ) R n n by 2 f ( x ) ij := 2 f x i x j ( x ) . Vris Cheung (University of Waterloo) CO 355 2010 6 / 22 Convex functions Outline 1 Preliminaries 2 Convex functions 3 Equivalent conditions of convexity Vris Cheung (University of Waterloo) CO 355 2010 7 / 22 Convex functions Convex functions Let S R n be convex (and nonempty). Definition A realvalued function f : S R is convex if for all x , y S , [ 0, 1 ] , f ( x + ( 1 ) y ) 6 f ( x ) + ( 1 ) f ( y ) . f : S R n R is concave if f is convex. Definition A function f : S R is strictly convex if for all distinct x , y S , ( 0, 1 ) , f ( x + ( 1 ) y ) < f ( x ) + ( 1 ) f ( y ) ....
View
Full
Document
This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.
 Winter '10
 Harvey

Click to edit the document details