Lecture13

Lecture13 - CO 355 Lecture 13 Convex functions Vris Cheung...

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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 . + Bolzano-Weierstrass 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 non-empty). Definition A real-valued 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 ) ....
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This note was uploaded on 06/16/2011 for the course CO 355 taught by Professor Harvey during the Winter '10 term at Waterloo.

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Lecture13 - CO 355 Lecture 13 Convex functions Vris Cheung...

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