Lecture-7 - Lecture-7 Step Length Selection Homework(Due 3.1 3.2 3.5 3.6 3.7 3.9 3.10 Show equation 3.44 The last step in the proof of Theorem

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1 Lecture-7 Step Length Selection Homework (Due 2/20/01) • 3.1 • 3.2 • 3.5 • 3.6 • 3.7 3.9 • 3.10 • Show equation 3.44 • The last step in the proof of Theorem 3.6. (see slides)
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2 Sufficient condition ) 1 , 0 ( , ) ( ) ( 1 1 + + c p f c x f p x f k T k k k k a a ) 1 , 0 ( , ) ( ) ( 1 1 - + c p f c x f p x f k T k k k k a a The reduction should be proportional to both the step length, and directional derivative. St line ) 1 , 0 ( , ) ( ) ( 1 1 + + c p f c x f p x f k T k k k k a a ) ( ) ( a a l p x f k k + 4 1 10 - = c Sufficient condition ) ( ) ( a a l p x f k k + Problem: The sufficient decrease condition is satisfied for all small values of step length
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3 Curvature condition ) 1 , ( , ) ( ) ( 1 2 2 c c p x f c p p x f k k T k k T k k + a The slope of is greater than times the gradient . ) 0 ( f ) ( k a f 2 c Derivative ) ( k a f gradient conjugate 1for . Newton - Quasi and Newton for 9 . 2 2 = = c c Curvature condition If the slope is strongly negative, that means we can reduce f further along the chosen direction If the slope is positive, it indicates we can not decrease f further in this direction.
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Wolfe conditions ) 1 , ( , ) ( ) ( 1 2 2 c c p x f c p p x f k k T k k T k k + a ) 1 , 0 ( , ) ( ) ( 1 1 + + c p f c x f p x f k T k k k k a a Sufficient decrease Curvature Backtracking Line Search with Terminate ) ( ; ) ( ) ( until ; set ); 1 , 0 ( , , 0 Choose a a ra a a a a a r a = + + k k T k k k k repeat end p f c x f p x f repeat c If line search method chooses its step length appropriately, we can dispense with the second condition This ensures that the step length is short enough to satisfy the sufficient decrease condition, but not too short. Newton
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This note was uploaded on 06/12/2011 for the course COT 6505 taught by Professor Shah during the Spring '07 term at University of Central Florida.

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Lecture-7 - Lecture-7 Step Length Selection Homework(Due 3.1 3.2 3.5 3.6 3.7 3.9 3.10 Show equation 3.44 The last step in the proof of Theorem

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