L5_func_more - Lecture 5 Verifying Convexity Operations...

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Lecture 5: Verifying Convexity Operations Preserving Convex Functions January 31, 2007
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Lecture 5 Outline I Verifying Convexity of a Function I Operations on Functions Preserving Convexity I Log-Transformation I Jensen’s Inequality I Extended-Value Functions I Epigraph and Level Sets Convex Optimization 1
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Lecture 5 Verifying Convexity of a Function We can verify that a given function f is convex by I Using the definition I Applying some special criteria Second-order conditions First-order conditions Reduction to a scalar function I Showing that f is obtained through operations preserving convexity Convex Optimization 2
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Lecture 5 First-Order Condition f is differentiable if dom ( f ) is open and the gradient f ( x ) = ± ∂f ( x ) ∂x 1 , ∂f ( x ) ∂x 2 , . . . , ∂f ( x ) ∂x n ! exists at each x dom f 1st-order condition: differentiable f is convex if and only if its domain is convex and f ( x ) + f ( x ) T ( z - x ) f ( z ) for all x, z dom ( f ) A first order approximation is a global underestimate of f Very important property used in algorithm designs and performance analysis Convex Optimization 3
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Lecture 5 Restriction of a convex function to a line f is convex if and only if dom f is convex and the function g : R 7→ R , g ( t ) = f ( x + tv ) , dom g = { t | x + tv dom ( f ) } is convex (in t ) for any x dom f , v
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L5_func_more - Lecture 5 Verifying Convexity Operations...

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