L25_varmet - Lecture 25: Efficiency of Proximal Bundle...

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Unformatted text preview: Lecture 25: Efficiency of Proximal Bundle Method Variable Metric Subgradient Methods April 23, 2007 Lecture 25 Outline Computing-Subgradients Simplified Proximal Bundle Method Convergence Rate of the Method Subdifferential of a Composite Function Some issues in optimization Convex Optimization 1 Lecture 25 Comment on-Subgradient Let f be a convex function with dom f and bounded subgradients over a set X . Let x X and > Let y X be such that k y- x k 2 L with L = max {k s k | s f ( z ) , z X } Any vector s f ( x ) is an-subgradient of f at y , i.e., f ( x ) f ( y ) Proof : We have for any z R n , f ( y ) + s T ( z- y ) = f ( y )- f ( x ) + f ( x ) + s T ( z- x + x- y ) f ( x ) + s T ( z- x ) + [ f ( y )- f ( x )] + k s kk x- y k f ( z ) + [ f ( y )- f ( x )] + L k x- y k Convex Optimization 2 Lecture 25 where the last inequality follows from s f ( x ) . By convexity of f , we have f ( y )- f ( x ) g T ( y- x ) for some g f ( y ) By subgradient boundedness, it follows that f ( y )- f ( x ) L k x- y k . Hence, for all z R n f ( y ) + s T ( z- y ) f ( z ) + 2 L k x- y k f ( z ) + showing that s f ( y ) Convex Optimization 3 Lecture 25-Subgradient for Dual Function q ( ) = inf x X { f ( x ) + T g ( x ) } Suppose x X attains the dual value q ( ) with an error > , i.e., x X is such that q ( ) + f ( x ) + T g ( x ) (1) Then g ( x ) q ( ) Proof : We need to show that q ( ) + g ( x ) T ( - ) q ( )- for any dom q . Using relation (1), we obtain q ( ) + g ( x ) T ( - ) f ( x ) + g ( x ) T - + g ( x ) T ( - ) = f ( x ) + T g ( x )- inf x X { f ( x ) + T g ( x ) } - = q ( )- Convex Optimization 4 Lecture 25 A Proximal Bundle Method Using Subgradients This method is also given by Kiwiel [2000]. It is an adaptation of the proximal method of Kiwiel proposed in 1995 What has been improved: Subgradients are used instead of-subgradients Stopping rule is based on actual error in the linearized model Scheme for bundle update is simpler [does not use error updates] Scheme for weight update is simpler [no need for a factor 10] As a result: There are fewer parameters to choose The overall procedure is simpler [less computationally demanding] The algorithm is easier for analysis Convex Optimization 5 Lecture 25...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.

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L25_varmet - Lecture 25: Efficiency of Proximal Bundle...

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