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

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Lecture 25: Efficiency of Proximal Bundle Method Variable Metric Subgradient Methods April 23, 2007

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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 > 0 Let y X be such that y - x 2 L with L = max { s | 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 )] + s x - y f ( z ) + [ f ( y ) - f ( x )] + L x - y Convex Optimization 2

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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 x - y . Hence, for all z R n f ( y ) + s T ( z - y ) f ( z ) + 2 L x - y f ( z ) + showing that s ∂ f ( y ) Convex Optimization 3
Lecture 25 -Subgradient for Dual Function q μ ) = inf x X { f ( x ) + ˜ μ T g ( x ) } Suppose ˜ x X attains the dual value q μ ) with an error > 0 , i.e., ˜ x X is such that q μ ) + f x ) + ˜ μ T g x ) (1) Then g x ) ∂ q μ ) Proof : We need to show that q μ ) + g x ) T ( μ - ˜ μ ) q ( μ ) - for any μ dom q . Using relation (1), we obtain q μ ) + g x ) T ( μ - ˜ μ ) f x ) + g x ) T ˜ μ - + g x ) T ( μ - ˜ μ ) = f x ) + μ T g x ) - inf x X { f ( x ) + μ T g ( x ) } - = q ( μ ) - Convex Optimization 4

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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 The Method Parameters Consider unconstrained minimization of convex f with dom f = R n The paper discusses a more general optimization subject to a convex and closed set X Parameters Minimal and maximal weight u min and u

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• Spring '07
• AngeliaNedich
• Mathematical optimization, Convex function, Convex Optimization, Convex analysis, Subgradient method

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L25_varmet - Lecture 25 Efficiency of Proximal Bundle...

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