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Unformatted text preview: Lecture 21: Subgradient Methods April 9, 2007 Lecture 21 Outline • Subgradients and Level Sets • Subgradient Method • Convergence and Convergence Rate Convex Optimization 1 Lecture 21 Subgradients and Level Sets A vector s is a subgradient of a convex function f : R n 7→ R at ˆ x ∈ dom f when subgradient inequality holds f ( z ) ≥ f (ˆ x ) + s T ( z ˆ x ) for all z ∈ dom f • We have interpreted the subgradient inequality in terms of a hyperplane in R n +1 supporting the epigraph epi f at (ˆ x,f (ˆ x )) Convex Optimization 2 Lecture 21 • Projecting the hyperplane and the epigraph on the set { ( x,w )  x ∈ R n , w = f (ˆ x ) } • The subgradient inequality can be interpreted in terms of a hyperplane in R n supporting the level set L γ ( f ) at ˆ x for γ = f (ˆ x ) • For z in the level set L f (ˆ x ) ( f ) = { z  f ( z ) ≤ f (ˆ x ) } , the subgradient inequality implies s T ( z ˆ x ) ≤ for all z ∈ L f (ˆ x ) ( f ) and the inequality is tight at z = ˆ x . Convex Optimization 3 Lecture 21 Another Interpretation of Subgradients • A subgradient of f at any ˆ x defines a hyperplane that cuts R n in two regions and one of them contains the level sets L γ ( f ) for all γ ≤ f (ˆ x ) • The optimal set X * lies entirely on one side of each such hyperplane • Cuttingplane method can be interpreted as successively “reducing” the size of a region where the “optimal solutions” are located, by increasing the number of hyperplanes that “cut” the space R n Convex Optimization 4 Lecture 21 Subgradient Method Constrained Convex Minimization minimize f ( x ) subject to x ∈ X General Assumption • The function f ( x ) is convex and dom f = R n • The set X is nonempty closed and convex Subgradient Method • Having the current iterate x k , a subgradient s k of f at x k , and a stepsize value α k > , the next iterate is given by x k +1 = P X [ x k α k s k ] • Polyak 1969, Ermoliev 1969, Shor 1985 Convex Optimization 5 Lecture 21 Property “ensuring” that subgradient methods work Along the negative subgradient direction...
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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.
 Spring '07
 AngeliaNedich

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