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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 • BundleType Methods using “Steepest Descent Idea” •Subgradients andSubdifferentals 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 α ↓ f ( x + αd ) f ( x ) α ≥ lim α ↓ s T ( x + αd x ) α = 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 α> f ( x + αd ) f ( x ) α ≤ 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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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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