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Unformatted text preview:  f ( x k + k d k ) T d k  c 2  f ( x k ) T d k  implies the curvature condition s T k y k , where s k = x k + 1x k and y k = f ( x k + 1 )  f ( x k ) . The second strong Wolfe condition implies that f ( x k + k d k ) T d k c 2  f ( x k ) T d k  = c 2 f ( x k ) T d k , since d k is a descent direction. (Note that this is just the regular second Wolfe condition.) Therefore, by subtracting the same term to both sides, we see that f ( x k + k d k ) T d k f ( x k ) T d k ( c 21 ) f ( x k ) T d k > 0. (1) Recall that k d k = x k + 1x k , multiply both sides of (1) by k and substitute to get f ( x k + 1 ) T ( x k + 1x k )  f ( x k ) T ( x k + 1x k ) > 0 which is equivalent to y T k s k > 0 . Problem 0 Page 2...
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 Spring '08
 Linderoth

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