L23_bundle

L23_bundle - Lecture 23: Steepest Descent Subgradient...

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Unformatted text preview: Lecture 23: Steepest Descent Subgradient Methods April 16, 2007 Lecture 23 Outline Directional Derivatives and Subgradients More Subgradient Properties Steepest Descent Subgradient Bundle-Type Methods using Steepest Descent Idea -Subgradients and-Subdifferentals Convex Optimization 1 Lecture 23 From Subdifferential to Directional Derivative Theorem Let f be convex with dom f = R n . We then have for any x R n and any d R n , f ( x ; d ) = max s f ( x ) s T d If we know the whole subdifferential f ( x ) , then for any direction d , the directional derivative at x is obtained by maximizing d T s over s f ( x ) There is a relation in the other direction: having the directional deriva- tives f ( x ; d ) for all d , one can recover the subdifferential f ( x ) Convex Optimization 2 Lecture 23 From Directional Derivative to Subdifferential Theorem Let f be convex with dom f = R n . We then have for any x R n and any d R n , f ( x ) = { s | f ( x ; d ) s T d for all d R n } (= K ) Proof : We have [from the definitions of subgradient and f ( x ; d ) ] that for any s f ( x ) , f ( x ; d ) = lim f ( x + d )- f ( x ) lim s T ( x + d- x ) = s T d Hence f ( x ) K . Suppose now s K , so that s T d f ( x ; d ) for all d R n . Thus, we have for any d R n : s T d inf > f ( x + d )- f ( x ) f ( x + d )- f ( x ) By letting d = z- x for any z R n , it follows that s f ( x ) Convex Optimization 3 Lecture 23 Directional Derivative and Optimality Theorem Optimality Condition Let f be convex with dom f = R n and let X R n be closed and convex. The vector x * is a minimizer of f over X if and only if f ( x * ; ( x- x * )) for all x X Proof : The result follows from the following two facts x * is optimal if and only if there is s f ( x * ) such that s T ( x- x * ) for all x X f ( x * ; d ) = max s f ( x * ) s...
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L23_bundle - Lecture 23: Steepest Descent Subgradient...

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