L22_dynrule - Lecture 22 Subgradient Methods Lecture 22...

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Lecture 22: Subgradient Methods April 11, 2007
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Lecture 22 Outline Methods with Dynamic Stepsize Rule Directional Derivatives and Subgradients -Subgradients and -Subdifferentals Convex Optimization 1
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Lecture 22 Method with Dynamic Stepsize with Known f * General Assumption The function f ( x ) is convex and dom f = R n The set X is nonempty closed and convex Theorem Assume that General Assumption holds and X * is nonempty. Then, { x k } generated by the subgradient method with the stepsize α k = f ( x k ) - f * s k 2 converges to an optimal solution. Convex Optimization 2
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Lecture 22 Proof We use the Basic Iterate Relation : for any y X and any k 0 , x k +1 - y 2 x k - y 2 - 2 α k ( f ( x k ) - f ( y )) + α 2 k s k 2 Letting y = x * for any x * X * , we obtain x k +1 - x * 2 x k - x * 2 - 2 α k ( f ( x k ) - f * ) + α 2 k s k 2 = x k - x * 2 - α k s k 2 2 f ( x k ) - f * s k 2 - α k = x k - x * 2 - ( f ( x k ) - f * ) 2 s k 2 The sequence { x k } is bounded, thus subgradient norms s k are uniformly bounded by some scalar L . Therefore, for all k and any x * X * , x k +1 - x * 2 x k - x * 2 - ( f ( x k ) - f * ) 2 L 2 (1) Convex Optimization 3
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Lecture 22 Eq. (1) implies that ( f ( x k ) - f * ) 2 L 2 x k - x * 2 - x k +1 - x * 2 , and k =0 ( f ( x k ) - f * ) 2 L 2 x 0 - x * 2 Hence f ( x k ) f * . Since { x k } ⊆ X is bounded, it has limit points that belong to X by closedness of X . By continuity of f and the fact f ( x k ) f * , it follows that all limit points are optimal. We now show that { x k } has a unique limit point. To arrive at a contradic- tion, suppose there are at least two: say ˜ x and ˆ x with ˜ x = ˆ x . Let { x m } M and { x k } K be subsequences converging to ˜ x and ˆ x , respectively. By Eq. (1), it follows that for all m, k with m k , x m - x * < x k - x * for all x * X * Let x * = ˆ x , and m ∈ M and k ∈ K . Then by taking limits, we obtain lim m →∞ m ∈M x m - ˆ x lim k →∞ k ∈K x k - ˆ x implying that ˜ x - ˆ x = 0 - a contradiction. Convex Optimization 4
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Lecture 22 Convergence Rate The convergence rate of the method with dynamic stepsize using f * is linear at best For a function f with sharp minima , i.e., such that for some η > 0 f ( x ) - f * η dist ( x, X * ) for all x The rate is linear x k - ˜ x * c k x 0 - ˜ x * for all k 0 where ˜ x * X * is the limit point of { x k } and
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