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Lecture13

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

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CO 355 Lecture 13 Convex functions Vris Cheung (Substitute class) October 26, 2010 Vris Cheung (University of Waterloo) CO 355 2010 1 / 22

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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

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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

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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

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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 , λ [ 0, 1 ] , f ( λ x + ( 1 - λ ) y ) 6 λ f ( x ) + ( 1 - λ ) 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 , λ ( 0, 1 ) , f ( λ x + ( 1 - λ ) y ) < λ f ( x ) + ( 1 - λ ) f ( y ) . Vris Cheung (University of Waterloo) CO 355 2010 8 / 22
Convex functions Convex functions The secant line lies above the curve through two points in S . Vris Cheung (University of Waterloo) CO 355 2010 9 / 22

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Convex functions Convex functions The secant line lies above the curve through two points in S . Vris Cheung (University of Waterloo) CO 355 2010 9 / 22
Convex functions Convex functions The secant line lies above the curve through two points in S .

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

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